Diskriminant - Vikipedi

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düzenlemeler hakkında tartışma [n]" accesskey="n"><span>Mesaj</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="İçindekiler" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">İçindekiler</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">kenar çubuğuna taşı</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">gizle</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Giriş</div> </a> </li> <li id="toc-İkinci_derecede_polinom" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#İkinci_derecede_polinom"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>İkinci derecede polinom</span> </div> </a> <button aria-controls="toc-İkinci_derecede_polinom-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>İkinci derecede polinom alt bölümünü aç/kapa</span> </button> <ul id="toc-İkinci_derecede_polinom-sublist" class="vector-toc-list"> <li id="toc-Gerçel_iki_köklü_tek_bilinmeyenli_ikinci_derecede_polinom_denklemin_çözülmesi" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gerçel_iki_köklü_tek_bilinmeyenli_ikinci_derecede_polinom_denklemin_çözülmesi"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Gerçel iki köklü tek bilinmeyenli ikinci derecede polinom denklemin çözülmesi</span> </div> </a> <ul id="toc-Gerçel_iki_köklü_tek_bilinmeyenli_ikinci_derecede_polinom_denklemin_çözülmesi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kompleks_iki_köklü_tek_bilinmeyenli_ikinci_derecede_polinom_denklemin_çözülmesi" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kompleks_iki_köklü_tek_bilinmeyenli_ikinci_derecede_polinom_denklemin_çözülmesi"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Kompleks iki köklü tek bilinmeyenli ikinci derecede polinom denklemin çözülmesi</span> </div> </a> <ul id="toc-Kompleks_iki_köklü_tek_bilinmeyenli_ikinci_derecede_polinom_denklemin_çözülmesi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kısaltılmış_diskriminant" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kısaltılmış_diskriminant"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Kısaltılmış diskriminant</span> </div> </a> <ul id="toc-Kısaltılmış_diskriminant-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Örnekler" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Örnekler"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Örnekler</span> </div> </a> <ul id="toc-Örnekler-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-İkinci_boyutta_kuadratik_formlar" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#İkinci_boyutta_kuadratik_formlar"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>İkinci boyutta kuadratik formlar</span> </div> </a> <ul id="toc-İkinci_boyutta_kuadratik_formlar-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Herhangi_bir_derecede_polinom" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Herhangi_bir_derecede_polinom"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Herhangi bir derecede polinom</span> </div> </a> <button aria-controls="toc-Herhangi_bir_derecede_polinom-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Herhangi bir derecede polinom alt bölümünü aç/kapa</span> </button> <ul id="toc-Herhangi_bir_derecede_polinom-sublist" class="vector-toc-list"> <li id="toc-Örnekler_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Örnekler_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Örnekler</span> </div> </a> <ul id="toc-Örnekler_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Genel_şekilde_ifade" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Genel_şekilde_ifade"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Genel şekilde ifade</span> </div> </a> <ul id="toc-Genel_şekilde_ifade-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Diskriminant_cebirsel_tam_sayılar_halkası" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Diskriminant_cebirsel_tam_sayılar_halkası"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Diskriminant cebirsel tam sayılar halkası</span> </div> </a> <ul id="toc-Diskriminant_cebirsel_tam_sayılar_halkası-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kaynakça" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kaynakça"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Kaynakça</span> </div> </a> <ul id="toc-Kaynakça-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dış_bağlantılar" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dış_bağlantılar"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Dış bağlantılar</span> </div> </a> <ul id="toc-Dış_bağlantılar-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="İçindekiler" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="İçindekiler tablosunu değiştir" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">İçindekiler tablosunu değiştir</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Diskriminant</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Başka bir dildeki sayfaya gidin. 43 dilde mevcut" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-43" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">43 dil</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%85%D9%8A%D8%B2" title="مميز - Arapça" lang="ar" hreflang="ar" data-title="مميز" data-language-autonym="العربية" data-language-local-name="Arapça" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%94%D1%8B%D1%81%D0%BA%D1%80%D1%8B%D0%BC%D1%96%D0%BD%D0%B0%D0%BD%D1%82" title="Дыскрымінант - Belarusça" lang="be" hreflang="be" data-title="Дыскрымінант" data-language-autonym="Беларуская" data-language-local-name="Belarusça" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA%D1%80%D0%B8%D0%BC%D0%B8%D0%BD%D0%B0%D0%BD%D1%82%D0%B0" title="Дискриминанта - Bulgarca" lang="bg" hreflang="bg" data-title="Дискриминанта" data-language-autonym="Български" data-language-local-name="Bulgarca" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Discriminant" title="Discriminant - Katalanca" lang="ca" hreflang="ca" data-title="Discriminant" data-language-autonym="Català" data-language-local-name="Katalanca" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Diskriminant" title="Diskriminant - Çekçe" lang="cs" hreflang="cs" data-title="Diskriminant" data-language-autonym="Čeština" data-language-local-name="Çekçe" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA%D1%80%D0%B8%D0%BC%D0%B8%D0%BD%D0%B0%D0%BD%D1%82" title="Дискриминант - Çuvaşça" lang="cv" hreflang="cv" data-title="Дискриминант" data-language-autonym="Чӑвашла" data-language-local-name="Çuvaşça" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Diskriminant" title="Diskriminant - Danca" lang="da" hreflang="da" data-title="Diskriminant" data-language-autonym="Dansk" data-language-local-name="Danca" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Diskriminante" title="Diskriminante - Almanca" lang="de" hreflang="de" data-title="Diskriminante" data-language-autonym="Deutsch" data-language-local-name="Almanca" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Discriminant" title="Discriminant - İngilizce" lang="en" hreflang="en" data-title="Discriminant" data-language-autonym="English" data-language-local-name="İngilizce" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Diskriminanto" title="Diskriminanto - Esperanto" lang="eo" hreflang="eo" data-title="Diskriminanto" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Discriminante" title="Discriminante - İspanyolca" lang="es" hreflang="es" data-title="Discriminante" data-language-autonym="Español" data-language-local-name="İspanyolca" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%A8%DB%8C%D9%86_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="مبین (ریاضیات) - Farsça" lang="fa" hreflang="fa" data-title="مبین (ریاضیات)" data-language-autonym="فارسی" data-language-local-name="Farsça" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Diskriminantti" title="Diskriminantti - Fince" lang="fi" hreflang="fi" data-title="Diskriminantti" data-language-autonym="Suomi" data-language-local-name="Fince" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Discriminant" title="Discriminant - Fransızca" lang="fr" hreflang="fr" data-title="Discriminant" data-language-autonym="Français" data-language-local-name="Fransızca" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Idirdheala%C3%AD" title="Idirdhealaí - İrlandaca" lang="ga" hreflang="ga" data-title="Idirdhealaí" data-language-autonym="Gaeilge" data-language-local-name="İrlandaca" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%93%D7%99%D7%A1%D7%A7%D7%A8%D7%99%D7%9E%D7%99%D7%A0%D7%A0%D7%98%D7%94" title="דיסקרימיננטה - İbranice" lang="he" hreflang="he" data-title="דיסקרימיננטה" data-language-autonym="עברית" data-language-local-name="İbranice" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Diszkrimin%C3%A1ns" title="Diszkrimináns - Macarca" lang="hu" hreflang="hu" data-title="Diszkrimináns" data-language-autonym="Magyar" data-language-local-name="Macarca" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B4%D5%AB%D5%BD%D5%AF%D6%80%D5%AB%D5%B4%D5%AB%D5%B6%D5%A1%D5%B6%D5%BF" title="Դիսկրիմինանտ - Ermenice" lang="hy" hreflang="hy" data-title="Դիսկրիմինանտ" data-language-autonym="Հայերեն" data-language-local-name="Ermenice" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Diskriminan" title="Diskriminan - Endonezce" lang="id" hreflang="id" data-title="Diskriminan" data-language-autonym="Bahasa Indonesia" data-language-local-name="Endonezce" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Diskriminanto" title="Diskriminanto - Ido" lang="io" hreflang="io" data-title="Diskriminanto" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Discriminante" title="Discriminante - İtalyanca" lang="it" hreflang="it" data-title="Discriminante" data-language-autonym="İtaliano" data-language-local-name="İtalyanca" class="interlanguage-link-target"><span>İtaliano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%88%A4%E5%88%A5%E5%BC%8F" title="判別式 - Japonca" lang="ja" hreflang="ja" data-title="判別式" data-language-autonym="日本語" data-language-local-name="Japonca" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA%D1%80%D0%B8%D0%BC%D0%B8%D0%BD%D0%B0%D0%BD%D1%82" title="Дискриминант - Kazakça" lang="kk" hreflang="kk" data-title="Дискриминант" data-language-autonym="Қазақша" data-language-local-name="Kazakça" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8C%90%EB%B3%84%EC%8B%9D" title="판별식 - Korece" lang="ko" hreflang="ko" data-title="판별식" data-language-autonym="한국어" data-language-local-name="Korece" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Diskriminantas" title="Diskriminantas - Litvanca" lang="lt" hreflang="lt" data-title="Diskriminantas" data-language-autonym="Lietuvių" data-language-local-name="Litvanca" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Diskriminants" title="Diskriminants - Letonca" lang="lv" hreflang="lv" data-title="Diskriminants" data-language-autonym="Latviešu" data-language-local-name="Letonca" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Pembeza_layan" title="Pembeza layan - Malayca" lang="ms" hreflang="ms" data-title="Pembeza layan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malayca" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Discriminant" title="Discriminant - Felemenkçe" lang="nl" hreflang="nl" data-title="Discriminant" data-language-autonym="Nederlands" data-language-local-name="Felemenkçe" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Diskriminant" title="Diskriminant - Norveççe Nynorsk" lang="nn" hreflang="nn" data-title="Diskriminant" data-language-autonym="Norsk nynorsk" data-language-local-name="Norveççe Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Diskriminant" title="Diskriminant - Norveççe Bokmål" lang="nb" hreflang="nb" data-title="Diskriminant" data-language-autonym="Norsk bokmål" data-language-local-name="Norveççe Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wyr%C3%B3%C5%BCnik_wielomianu" title="Wyróżnik wielomianu - Lehçe" lang="pl" hreflang="pl" data-title="Wyróżnik wielomianu" data-language-autonym="Polski" data-language-local-name="Lehçe" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA%D1%80%D0%B8%D0%BC%D0%B8%D0%BD%D0%B0%D0%BD%D1%82" title="Дискриминант - Rusça" lang="ru" hreflang="ru" data-title="Дискриминант" data-language-autonym="Русский" data-language-local-name="Rusça" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA%D1%80%D0%B8%D0%BC%D0%B8%D0%BD%D0%B0%D0%BD%D1%82" title="Дискриминант - Yakutça" lang="sah" hreflang="sah" data-title="Дискриминант" data-language-autonym="Саха тыла" data-language-local-name="Yakutça" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Discriminant" title="Discriminant - Simple English" lang="en-simple" hreflang="en-simple" data-title="Discriminant" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Diskriminant_(matematika)" title="Diskriminant (matematika) - Slovakça" lang="sk" hreflang="sk" data-title="Diskriminant (matematika)" data-language-autonym="Slovenčina" data-language-local-name="Slovakça" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Diskriminanta" title="Diskriminanta - Slovence" lang="sl" hreflang="sl" data-title="Diskriminanta" data-language-autonym="Slovenščina" data-language-local-name="Slovence" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Dallori" title="Dallori - Arnavutça" lang="sq" hreflang="sq" data-title="Dallori" data-language-autonym="Shqip" data-language-local-name="Arnavutça" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AE%A9%E0%AF%8D%E0%AE%AE%E0%AF%88%E0%AE%95%E0%AE%BE%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%BF" title="தன்மைகாட்டி - Tamilce" lang="ta" hreflang="ta" data-title="தன்மைகாட்டி" data-language-autonym="தமிழ்" data-language-local-name="Tamilce" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%94%E0%B8%B4%E0%B8%AA%E0%B8%84%E0%B8%A3%E0%B8%B4%E0%B8%A1%E0%B8%B4%E0%B9%81%E0%B8%99%E0%B8%99%E0%B8%95%E0%B9%8C" title="ดิสคริมิแนนต์ - Tayca" lang="th" hreflang="th" data-title="ดิสคริมิแนนต์" data-language-autonym="ไทย" data-language-local-name="Tayca" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA%D1%80%D0%B8%D0%BC%D1%96%D0%BD%D0%B0%D0%BD%D1%82" title="Дискримінант - Ukraynaca" lang="uk" hreflang="uk" data-title="Дискримінант" data-language-autonym="Українська" data-language-local-name="Ukraynaca" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Diskriminant" title="Diskriminant - Özbekçe" lang="uz" hreflang="uz" data-title="Diskriminant" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Özbekçe" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Bi%E1%BB%87t_th%E1%BB%A9c" title="Biệt thức - Vietnamca" lang="vi" hreflang="vi" data-title="Biệt thức" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamca" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%88%A4%E5%88%AB%E5%BC%8F" title="判别式 - Çince" lang="zh" hreflang="zh" data-title="判别式" data-language-autonym="中文" data-language-local-name="Çince" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q192487#sitelinks-wikipedia" title="Dillerarası bağlantıları değiştir" class="wbc-editpage">Bağlantıları değiştir</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Ad alanları"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Diskriminant" title="İçerik sayfasını göster [c]" accesskey="c"><span>Madde</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Tart%C4%B1%C5%9Fma:Diskriminant" rel="discussion" title="İçerik ile ilgili tartışma [t]" accesskey="t"><span>Tartışma</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Dil varyantını değiştir" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Türkçe</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Görünüm"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Diskriminant"><span>Oku</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Diskriminant&veaction=edit" title="Bu sayfayı değiştir [v]" accesskey="v"><span>Değiştir</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Diskriminant&action=edit" title="Bu sayfanın kaynak kodunu düzenleyin [e]" accesskey="e"><span>Kaynağı değiştir</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Diskriminant&action=history" title="Bu sayfanın geçmiş sürümleri [h]" accesskey="h"><span>Geçmişi gör</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Sayfa araçları"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Araçlar" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Araçlar</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Araçlar</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">kenar çubuğuna taşı</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">gizle</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Daha fazla seçenek" > <div class="vector-menu-heading"> Eylemler </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Diskriminant"><span>Oku</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Diskriminant&veaction=edit" title="Bu sayfayı değiştir [v]" accesskey="v"><span>Değiştir</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Diskriminant&action=edit" title="Bu sayfanın kaynak kodunu düzenleyin [e]" accesskey="e"><span>Kaynağı değiştir</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Diskriminant&action=history"><span>Geçmişi gör</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Genel </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%C3%96zel:SayfayaBa%C4%9Flant%C4%B1lar/Diskriminant" title="Bu sayfaya bağlantı vermiş tüm viki sayfalarının listesi [j]" accesskey="j"><span>Sayfaya bağlantılar</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%C3%96zel:%C4%B0lgiliDe%C4%9Fi%C5%9Fiklikler/Diskriminant" rel="nofollow" title="Bu sayfadan bağlantı verilen sayfalardaki son değişiklikler [k]" accesskey="k"><span>İlgili değişiklikler</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%C3%96zel:%C3%96zelSayfalar" title="Tüm özel sayfaların listesi [q]" accesskey="q"><span>Özel sayfalar</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Diskriminant&oldid=32761324" title="Bu sayfanın bu revizyonuna kalıcı bağlantı"><span>Kalıcı bağlantı</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Diskriminant&action=info" title="Bu sayfa hakkında daha fazla bilgi"><span>Sayfa bilgisi</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=%C3%96zel:KaynakG%C3%B6ster&page=Diskriminant&id=32761324&wpFormIdentifier=titleform" title="Bu sayfadan nasıl kaynak göstereceği hakkında bilgi"><span>Bu sayfayı kaynak göster</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=%C3%96zel:UrlShortener&url=https%3A%2F%2Ftr.wikipedia.org%2Fwiki%2FDiskriminant"><span>Kısaltılmış URL'yi al</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=%C3%96zel:QrCode&url=https%3A%2F%2Ftr.wikipedia.org%2Fwiki%2FDiskriminant"><span>Karekodu indir</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Yazdır/dışa aktar </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=%C3%96zel:Kitap&bookcmd=book_creator&referer=Diskriminant"><span>Bir kitap oluştur</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%C3%96zel:DownloadAsPdf&page=Diskriminant&action=show-download-screen"><span>PDF olarak indir</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Diskriminant&printable=yes" title="Bu sayfanın basılmaya uygun sürümü [p]" accesskey="p"><span>Basılmaya uygun görünüm</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Diğer projelerde </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q192487" title="Veri havuzundaki ilgili ögeye bağlantı [g]" accesskey="g"><span>Vikiveri ögesi</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Sayfa araçları"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Görünüm"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Görünüm</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">kenar çubuğuna taşı</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">gizle</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Vikipedi, özgür ansiklopedi</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="tr" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Dosya:Quadratic_eq_discriminant.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Quadratic_eq_discriminant.svg/250px-Quadratic_eq_discriminant.svg.png" decoding="async" width="250" height="214" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Quadratic_eq_discriminant.svg/375px-Quadratic_eq_discriminant.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/Quadratic_eq_discriminant.svg/500px-Quadratic_eq_discriminant.svg.png 2x" data-file-width="384" data-file-height="329" /></a><figcaption>Gerçel sayılı katsayıları olan ikinci derece denklemin köklerinin bulunması için hesaplanan diskriminant değerleri bileşimi</figcaption></figure> <p><b>Diskriminant</b> <a href="/wiki/Matematik" title="Matematik">matematik</a> biliminde bir <a href="/wiki/Cebir" title="Cebir">cebirsel</a> kavramdır. Gerçel katsayılı <a href="/w/index.php?title=%C4%B0kinci_derece_polinom_denklemler&action=edit&redlink=1" class="new" title="İkinci derece polinom denklemler (sayfa mevcut değil)">ikinci derece polinom denklemlerin</a> çözümü için kullanılır. İkinci dereceden büyük herhangi bir <a href="/wiki/Polinom" title="Polinom">polinomun</a> köklerinin bulunması için de bu kavram, <a href="/w/index.php?title=K%C3%B6klerin_toplam%C4%B1&action=edit&redlink=1" class="new" title="Köklerin toplamı (sayfa mevcut değil)">köklerin toplamı</a> için gereken ifadenin ve <a href="/w/index.php?title=K%C3%B6klerin_%C3%A7arp%C4%B1m%C4%B1&action=edit&redlink=1" class="new" title="Köklerin çarpımı (sayfa mevcut değil)">köklerin çarpımı</a> için gereken ifadenin bulunması suretiyle genişletilmiştir. Bir polinom için çoklu köklerin varlığı veya yokluğu için gereken koşul da diskriminantın varlığı ve yokluğu ile bulunabilmektedir. </p><p><b>Diskriminant</b> kavramı polinomların incelenmesinden daha başka matematik alanlarda da kullanılmaktadır. Bu kavramın kullanışı <a href="/wiki/Konik_kesit" class="mw-redirect" title="Konik kesit">konik kesitlerin</a> ve genel olarak kuadratik şekillerin daha iyi anlaşılmasına izin vermektedir. <a href="/wiki/Galois_teorisi" title="Galois teorisi">Galois teorisi</a>'nin <a href="/w/index.php?title=Kuadratik_formlar&action=edit&redlink=1" class="new" title="Kuadratik formlar (sayfa mevcut değil)">kuadratik formlara</a> veya <a href="/w/index.php?title=Say%C4%B1lar_sonlu_uzant%C4%B1s%C4%B1&action=edit&redlink=1" class="new" title="Sayılar sonlu uzantısı (sayfa mevcut değil)">sayılar sonlu uzantısı</a> hakkındaki gelişmelerde de <i>diskriminant</i> kavramı rol oynar. Matris sistemindeki <a href="/wiki/Determinant" title="Determinant">determinant</a> hesaplanmasının temelinde de <i>diskriminant</i> kavramı yatmaktadır. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="İkinci_derecede_polinom"><span id=".C4.B0kinci_derecede_polinom"></span>İkinci derecede polinom</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diskriminant&veaction=edit&section=1" title="Değiştirilen bölüm: İkinci derecede polinom" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Diskriminant&action=edit&section=1" title="Bölümün kaynak kodunu değiştir: İkinci derecede polinom"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Gerçel_iki_köklü_tek_bilinmeyenli_ikinci_derecede_polinom_denklemin_çözülmesi"><span id="Ger.C3.A7el_iki_k.C3.B6kl.C3.BC_tek_bilinmeyenli_ikinci_derecede_polinom_denklemin_.C3.A7.C3.B6z.C3.BClmesi"></span>Gerçel iki köklü tek bilinmeyenli ikinci derecede polinom denklemin çözülmesi</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diskriminant&veaction=edit&section=2" title="Değiştirilen bölüm: Gerçel iki köklü tek bilinmeyenli ikinci derecede polinom denklemin çözülmesi" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Diskriminant&action=edit&section=2" title="Bölümün kaynak kodunu değiştir: Gerçel iki köklü tek bilinmeyenli ikinci derecede polinom denklemin çözülmesi"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tek bilinmeyenli ikinci derecede bir polinom denklem ele alalım ve denklemde <i>a</i>, <i>b</i> ve <i>c</i> üç gerçel sayılı <a href="/wiki/Katsay%C4%B1" title="Katsayı">katsayı</a> olsun ve <i>a</i> değeri <i>0</i> dan değişik olsun </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax^{2}+bx+c=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax^{2}+bx+c=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23e70cfa003f402d108ec04d97983fb62f69536e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.89ex; height:2.843ex;" alt="{\displaystyle ax^{2}+bx+c=0}"></span> </center> <p>denklemi ve a ≠ 0 olsun. </p><p>Bu tek bilinmeyenli ikinci derecede polinom denklemin <b>diskriminant</b>ı şöyle tanımlanan Δ (delta) sayısı ile ifade edilir: </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =b^{2}-4ac\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>a</mi> <mi>c</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =b^{2}-4ac\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cdb038b60dd75a2021e7eb37030efdf05861905" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.971ex; height:2.843ex;" alt="{\displaystyle \Delta =b^{2}-4ac\;}"></span></center> <p>Diskriminant'ın bilinmesi bu tek bilinmeyenli ikinci derece polinomun çözülmesini sağlar: </p><p>a) Δ > 0 yani Δ pozitif ise, denklemin farklı iki gerçel kökü vardır. <i>x</i><sub>1</sub> ve <i>x</i><sub>2</sub> olarak ifade edilen bu iki kök şu formül kullanılarak bulunur: </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}={\frac {-b+{\sqrt {\Delta }}}{2a}}\quad {\text{ ve}}\quad x_{2}={\frac {-b-{\sqrt {\Delta }}}{2a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>b</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi mathvariant="normal">Δ</mi> </msqrt> </mrow> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> ve</mtext> </mrow> <mspace width="1em" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>b</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi mathvariant="normal">Δ</mi> </msqrt> </mrow> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}={\frac {-b+{\sqrt {\Delta }}}{2a}}\quad {\text{ ve}}\quad x_{2}={\frac {-b-{\sqrt {\Delta }}}{2a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce7caa4dfb34386beec3abc1081265a93136c932" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:39.158ex; height:5.843ex;" alt="{\displaystyle x_{1}={\frac {-b+{\sqrt {\Delta }}}{2a}}\quad {\text{ ve}}\quad x_{2}={\frac {-b-{\sqrt {\Delta }}}{2a}}}"></span></center> <p>b) Δ = 0 yani Δ sıfıra eşit ise, denklemin, değerleri birbirleriyle çakışan, yani birbirine eşit, iki gerçel kökü vardır: </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax^{2}+bx+c=a(x+{\frac {b}{2a}})^{2}\quad {\text{et}}\quad x_{1}=x_{2}=-{\frac {b}{2a}}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>et</mtext> </mrow> <mspace width="1em" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax^{2}+bx+c=a(x+{\frac {b}{2a}})^{2}\quad {\text{et}}\quad x_{1}=x_{2}=-{\frac {b}{2a}}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52e1919ab8a0846aeadac4c461361428ca549c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:50.448ex; height:5.343ex;" alt="{\displaystyle ax^{2}+bx+c=a(x+{\frac {b}{2a}})^{2}\quad {\text{et}}\quad x_{1}=x_{2}=-{\frac {b}{2a}}\;}"></span></center> <p>c) Δ < 0 yani Δ negatif ise, denklemin gerçel kökü yoktur yani denklemin çözümü bulunamaz. </p> <div class="mw-heading mw-heading3"><h3 id="Kompleks_iki_köklü_tek_bilinmeyenli_ikinci_derecede_polinom_denklemin_çözülmesi"><span id="Kompleks_iki_k.C3.B6kl.C3.BC_tek_bilinmeyenli_ikinci_derecede_polinom_denklemin_.C3.A7.C3.B6z.C3.BClmesi"></span>Kompleks iki köklü tek bilinmeyenli ikinci derecede polinom denklemin çözülmesi</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diskriminant&veaction=edit&section=3" title="Değiştirilen bölüm: Kompleks iki köklü tek bilinmeyenli ikinci derecede polinom denklemin çözülmesi" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Diskriminant&action=edit&section=3" title="Bölümün kaynak kodunu değiştir: Kompleks iki köklü tek bilinmeyenli ikinci derecede polinom denklemin çözülmesi"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote navigation-not-searchable">Ana madde: <a href="/w/index.php?title=Kompleks_k%C3%B6klu_polinom_denklem&action=edit&redlink=1" class="new" title="Kompleks köklu polinom denklem (sayfa mevcut değil)">Kompleks köklu polinom denklem</a></div> <p>Eğer <i>a</i>, <i>b</i> ve <i>c</i> <a href="/wiki/Kompleks_say%C4%B1" class="mw-redirect" title="Kompleks sayı">kompleks (karmaşık) sayılar</a> ise veya denklemin çözümü için kompleks sayı kullanılması kabul edilmişse durum biraz daha değişiktir. <a href="/wiki/D%27Alembert-Gauss_teoremi" class="mw-redirect" title="D'Alembert-Gauss teoremi">D'Alembert-Gauss teoremine</a> göre denklemin en aşağı bir tane çözümünün bulunması gerekir. Kompleks sayılıların ise her zaman iki tane kare kökü bulunur; yani öyle bir δ değeri vardır ki bunun karesi (δ<sup>2</sup>) Δ'ya eşittir. Buna göre </p><p>a) Eğer <b>diskriminant</b> <i>sıfır</i> dan değişik bir değerde ise, denklemin iki çözüm değeri, yani <i>x</i><sub>1</sub> ve <i>x</i><sub>2</sub>, şu formülle bulunur: </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}={\frac {-b+\delta }{2a}}\quad {\text{ ve }}\quad x_{2}={\frac {-b-\delta }{2a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>b</mi> <mo>+</mo> <mi>δ</mi> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> ve </mtext> </mrow> <mspace width="1em" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>b</mi> <mo>−</mo> <mi>δ</mi> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}={\frac {-b+\delta }{2a}}\quad {\text{ ve }}\quad x_{2}={\frac {-b-\delta }{2a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0368525e605b2270e3d707def191fe98a8f55c5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.093ex; height:5.343ex;" alt="{\displaystyle x_{1}={\frac {-b+\delta }{2a}}\quad {\text{ ve }}\quad x_{2}={\frac {-b-\delta }{2a}}}"></span></center> <p>b) Eğer <b>diskriminant</b> değeri <i>sıfır</i> ise denklemin çözümü olarak birbiriyle çakışmış eşit şu iki tane kök <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}"></span> bulunur:<br /> </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}={\frac {-b}{2a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>b</mi> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}={\frac {-b}{2a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efc9ad91c7d8ae6c0d542fbcbd8f91dc61a49075" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.124ex; height:5.343ex;" alt="{\displaystyle x_{1}={\frac {-b}{2a}}}"></span></center> <div class="mw-heading mw-heading3"><h3 id="Kısaltılmış_diskriminant"><span id="K.C4.B1salt.C4.B1lm.C4.B1.C5.9F_diskriminant"></span>Kısaltılmış diskriminant</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diskriminant&veaction=edit&section=4" title="Değiştirilen bölüm: Kısaltılmış diskriminant" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Diskriminant&action=edit&section=4" title="Bölümün kaynak kodunu değiştir: Kısaltılmış diskriminant"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bazen ikinci derecedeki polinom denklem şu şekilde yazılmaktadır: </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax^{2}+2b'x+c=0\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>b</mi> <mo>′</mo> </msup> <mi>x</mi> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax^{2}+2b'x+c=0\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e306ecf8183ce71fbded631a8a07380c00d6ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.382ex; height:2.843ex;" alt="{\displaystyle ax^{2}+2b'x+c=0\;}"></span></center> <p>Bu şekilde değişik bir diskriminant bilinir ve bu <i>kısaltılmış diskriminant</i> (Δ') şöyle tanımlanır: </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta '=b'^{2}-ac\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Δ</mi> <mo>′</mo> </msup> <mo>=</mo> <msup> <mi>b</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mi>c</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta '=b'^{2}-ac\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aff9a2d4535d13099bf5aba10625fed99ab3b878" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.946ex; height:2.843ex;" alt="{\displaystyle \Delta '=b'^{2}-ac\;}"></span></center> <p>Eğer bu denklemin kökleri varsa, şöyle bulunurlar: </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}={\frac {-b'+\delta '}{a}}\quad {\text{et}}\quad x_{2}={\frac {-b'-\delta '}{a}}\quad {\text{avec}}\quad \delta '^{2}=\Delta '=b'^{2}-ac}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>+</mo> <msup> <mi>δ</mi> <mo>′</mo> </msup> </mrow> <mi>a</mi> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>et</mtext> </mrow> <mspace width="1em" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>−</mo> <msup> <mi>δ</mi> <mo>′</mo> </msup> </mrow> <mi>a</mi> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>avec</mtext> </mrow> <mspace width="1em" /> <msup> <mi>δ</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>=</mo> <msup> <mi mathvariant="normal">Δ</mi> <mo>′</mo> </msup> <mo>=</mo> <msup> <mi>b</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}={\frac {-b'+\delta '}{a}}\quad {\text{et}}\quad x_{2}={\frac {-b'-\delta '}{a}}\quad {\text{avec}}\quad \delta '^{2}=\Delta '=b'^{2}-ac}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/734bcf92b80918592c7f29dee6105236ca4c668e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:63.417ex; height:5.509ex;" alt="{\displaystyle x_{1}={\frac {-b'+\delta '}{a}}\quad {\text{et}}\quad x_{2}={\frac {-b'-\delta '}{a}}\quad {\text{avec}}\quad \delta '^{2}=\Delta '=b'^{2}-ac}"></span></center> <div class="mw-heading mw-heading3"><h3 id="Örnekler"><span id=".C3.96rnekler"></span>Örnekler</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diskriminant&veaction=edit&section=5" title="Değiştirilen bölüm: Örnekler" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Diskriminant&action=edit&section=5" title="Bölümün kaynak kodunu değiştir: Örnekler"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>a) İlk olarak şu örnek denklemin çözümünü arayalım: </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5x^{2}-5x+1=0\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>5</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5x^{2}-5x+1=0\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bfddd236600b2c766e368a68ce4356e076af90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.788ex; height:2.843ex;" alt="{\displaystyle 5x^{2}-5x+1=0\;}"></span></center> <p>Çözüm iki kök bulunmasını gerektirir. Bu iki kökün <i>x</i><sub>1</sub> ve <i>x</i><sub>2</sub> olduğunu kabul edelim. Bu iki kökü, yani <i>x</i><sub>1</sub> ve <i>x</i><sub>2</sub> çözüm değerlerini bulmak için, şu Δ diskriminant ifadesi incelenir ve bu diskriminant değeri kuadratik denklem çözüm formülüne konulup şu iki gerçel kök bulunur:: </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =(-5)^{2}-4\times 5\times 1=5\quad {\text{ ve }}\quad x_{1}={\frac {5+{\sqrt {5}}}{10}},\quad x_{2}={\frac {5-{\sqrt {5}}}{10}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>5</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mo>×</mo> <mn>5</mn> <mo>×</mo> <mn>1</mn> <mo>=</mo> <mn>5</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> ve </mtext> </mrow> <mspace width="1em" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>5</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>10</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>5</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>10</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =(-5)^{2}-4\times 5\times 1=5\quad {\text{ ve }}\quad x_{1}={\frac {5+{\sqrt {5}}}{10}},\quad x_{2}={\frac {5-{\sqrt {5}}}{10}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/499823337146964851095fe741806a7f932c0d01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:66.047ex; height:5.843ex;" alt="{\displaystyle \Delta =(-5)^{2}-4\times 5\times 1=5\quad {\text{ ve }}\quad x_{1}={\frac {5+{\sqrt {5}}}{10}},\quad x_{2}={\frac {5-{\sqrt {5}}}{10}}.}"></span></center> <p>b) İkinci örnek olarak verilen denklem şu olsun: </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+6x+9=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>6</mn> <mi>x</mi> <mo>+</mo> <mn>9</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+6x+9=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d88aa33fcd0be2e393956482381b66d4f2011fe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.98ex; height:2.843ex;" alt="{\displaystyle x^{2}+6x+9=0}"></span></center> <p>ve bunun diskriminant değeri sıfır olarak şöyle bulunur: </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =6^{2}-4(1)(9)=36-36=0\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <msup> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>9</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>36</mn> <mo>−</mo> <mn>36</mn> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =6^{2}-4(1)(9)=36-36=0\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4936e3252adbfb593bee7d044d2e4f5e449c6e0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.692ex; height:3.176ex;" alt="{\displaystyle \Delta =6^{2}-4(1)(9)=36-36=0\;}"></span></center> <p>Bu demektir ki bu denklem çözümü birbirine eşit iki gerçel kök olur </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+6x+9=(x+3)^{2}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>6</mn> <mi>x</mi> <mo>+</mo> <mn>9</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>3</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+6x+9=(x+3)^{2}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1349a0a46a04129de806e5055cf742a24448c32a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.659ex; height:3.176ex;" alt="{\displaystyle x^{2}+6x+9=(x+3)^{2}\;}"></span></center> <p>Bu birbirine çakışık iki kök değeri -3 olur. </p><p>c) Son olarak örnek denklem şu olsun: </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+x+1=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+x+1=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c5e55ad884c42814df1c0e098670f34d685d0eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.818ex; height:2.843ex;" alt="{\displaystyle x^{2}+x+1=0}"></span></center> <p>Bu denklem işin diskriminant Δ değeri şu olur: </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =1^{2}-4(1)(1)=-3\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>3</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =1^{2}-4(1)(1)=-3\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7792f7b20c344e832e0c646404122f400371fe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.912ex; height:3.176ex;" alt="{\displaystyle \Delta =1^{2}-4(1)(1)=-3\;}"></span></center> <p>yani Δ negatiftir. Bu halde denklemin gerçel sayılarla kökleri bulunmamaktadır. Fakat bu halde kompleks kökleri bulunabilir. Diskriminantın kare kökü <i>i</i>√3 olur ve burada <i>i</i> "sanal birim" operatörüdür. Bundan dolayı şu çözüm ortaya çıkar: </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \quad {\text{et}}\quad x_{1}=-{\frac {1}{2}}+i{\frac {\sqrt {3}}{2}},\quad x_{1}=-{\frac {1}{2}}-i{\frac {\sqrt {3}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>et</mtext> </mrow> <mspace width="1em" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \quad {\text{et}}\quad x_{1}=-{\frac {1}{2}}+i{\frac {\sqrt {3}}{2}},\quad x_{1}=-{\frac {1}{2}}-i{\frac {\sqrt {3}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c80b81787d031f8c34592c0eae9c13715e09d30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.252ex; height:5.843ex;" alt="{\displaystyle \ \quad {\text{et}}\quad x_{1}=-{\frac {1}{2}}+i{\frac {\sqrt {3}}{2}},\quad x_{1}=-{\frac {1}{2}}-i{\frac {\sqrt {3}}{2}}}"></span></center><p>. </p><div class="mw-heading mw-heading3"><h3 id="İkinci_boyutta_kuadratik_formlar"><span id=".C4.B0kinci_boyutta_kuadratik_formlar"></span>İkinci boyutta kuadratik formlar</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diskriminant&veaction=edit&section=6" title="Değiştirilen bölüm: İkinci boyutta kuadratik formlar" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Diskriminant&action=edit&section=6" title="Bölümün kaynak kodunu değiştir: İkinci boyutta kuadratik formlar"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote navigation-not-searchable">Ana madde: <a href="/w/index.php?title=Kuadratik_form&action=edit&redlink=1" class="new" title="Kuadratik form (sayfa mevcut değil)">Kuadratik form</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Dosya:Drini-conjugatehyperbolas.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Drini-conjugatehyperbolas.svg/280px-Drini-conjugatehyperbolas.svg.png" decoding="async" width="280" height="280" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Drini-conjugatehyperbolas.svg/420px-Drini-conjugatehyperbolas.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/Drini-conjugatehyperbolas.svg/560px-Drini-conjugatehyperbolas.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>eğer kuadratik formun diskriminantı negatif ise, φ(<i>x</i>, <i>y</i>) = <i>a</i> ile tanımlanan <i>R</i><sup>2</sup> noktaları ensamblı bir hiperboldur. Eğer <i>a</i> pozitif ise, mavi ile gösterilen eğriye benzer şekil alir. Eğer <i>a</i> negatif ise ortaya çıkan eğri yeşil eğri benzeridir. Eğer <i>a</i> sıfıra eşitse, hiperbol dejenere olur ve kırmızı eğri benzeri bir eğri oluşur.</figcaption></figure> <p><a href="/wiki/Reel_say%C4%B1lar" title="Reel sayılar">Gerçel sayılar</a> seti üzerinde, iki değişkenli (<i>x</i> ve <i>y</i>) iki boyutlu φ kuadratik formu şu formülle ifade edilir: </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (x,y)=ax^{2}+bxy+cy^{2}\quad {\text{ burada }}\quad a,b,c\in \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mi>c</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> burada </mtext> </mrow> <mspace width="1em" /> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (x,y)=ax^{2}+bxy+cy^{2}\quad {\text{ burada }}\quad a,b,c\in \mathbb {K} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1bd766a5a9a120562572a46ad9945cf40dbe0e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.817ex; height:3.176ex;" alt="{\displaystyle \varphi (x,y)=ax^{2}+bxy+cy^{2}\quad {\text{ burada }}\quad a,b,c\in \mathbb {K} }"></span></center> <p>Kuadratik form aynı zamanda bir <a href="/wiki/Matris" class="mw-disambig" title="Matris">matris</a> ifade ile de gösterilebilir: </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (x,y)={\begin{pmatrix}x&y\end{pmatrix}}{\begin{pmatrix}a&{\frac {b}{2}}\\{\frac {b}{2}}&c\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi>y</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mi>c</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (x,y)={\begin{pmatrix}x&y\end{pmatrix}}{\begin{pmatrix}a&{\frac {b}{2}}\\{\frac {b}{2}}&c\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae59be8bbc58fe57fb60fc9e53a5543e50829f99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:32.889ex; height:7.843ex;" alt="{\displaystyle \varphi (x,y)={\begin{pmatrix}x&y\end{pmatrix}}{\begin{pmatrix}a&{\frac {b}{2}}\\{\frac {b}{2}}&c\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}}"></span></center> <p>Bu matris şeklinde ifadenin <a href="/wiki/Determinant" title="Determinant">determinantinin</a> açılması, daha önce diskriminant için verilen ifadeye, yani -1/4(<i>b</i><sup>2</sup> - 4<i>ac</i>) ifadesine eşittir. Bir <a href="/w/index.php?title=Ge%C3%A7en_matris&action=edit&redlink=1" class="new" title="Geçen matris (sayfa mevcut değil)">geçen matris</a> <i>P</i> kullanarak yapılan bir baz değişmesi bu determinatın değerinde değişme yapar. Daha detaylı bir açıklama ile, yeni baz için değer eski baz ile <i>P</i> determinantının karesinin çarpımına eşittir ve determinantın işareti değişmeden aynı kalmaktadır. Bu analizin incelenmesi daha ayrıntılı bir maddede yapılmaktadır. </p><p>Bunun için iki boyutlu kuadratik formları için üç tane farklı tanımlama yapılmaktadır. <i>B</i> bazında olan kuadratik formun diskriminantı, <i>B</i> bazındaki kuadratik forma bağlı olan matrisin determinantı olur. Daha önceki hale benzer bir açıklama ve hesaplama ile kuadratik formun diskriminantının <i>b</i><sup>2</sup> - 4<i>ac</i> ifadesine eşit olduğu tanımlanabilir. Sonra, kuadratik formun determinantına bağlı tek değişmez gibi, diskriminant da +1, 0 veya -1 değerleri alabilen determinant işareti olarak tanımlanır. </p><p>Diskriminant kuadratik formları üç tane değişik gruba ayırmaktadır. İki boyutta, <a href="/w/index.php?title=Kanonik_baz&action=edit&redlink=1" class="new" title="Kanonik baz (sayfa mevcut değil)">kanonik bazda</a> determinatın değerinin diskriminantı tanımlaması yapıldıktan sonra, eğer verilmiş bir <i>a</i> değeri icin diskriminantın işareti pozitif ise, φ(<i>x</i>, <i>y</i>) = <i>a</i> değişebilirinin (<i>x</i>, <i>y</i>) noktalarının <i>E</i><sub>a</sub> ensamblı bir <a href="/wiki/Elips" title="Elips">elipse</a> karşıttır veya ensambl boştur. Eğer diskriminant sıfır ise, bu halde <i>E</i><sub>a</sub> bir <a href="/wiki/Parabol" title="Parabol">parabol</a>'a karşıt olur. Eğer diskriminant negatif ise, <i>E</i><sub>a</sub> bir <a href="/wiki/Hiperbol" title="Hiperbol">hiperbol</a> olur. Kuadratik formlar üç farklı şekilde <a href="/w/index.php?title=Konik_seksiyon&action=edit&redlink=1" class="new" title="Konik seksiyon (sayfa mevcut değil)">konik seksiyon</a> elde etmeye izin verir. </p> <div class="mw-heading mw-heading2"><h2 id="Herhangi_bir_derecede_polinom">Herhangi bir derecede polinom</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diskriminant&veaction=edit&section=7" title="Değiştirilen bölüm: Herhangi bir derecede polinom" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Diskriminant&action=edit&section=7" title="Bölümün kaynak kodunu değiştir: Herhangi bir derecede polinom"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bir polinom için kök değerini diskriminant yardımı ile çıkarma yöntemi ikiden büyük polinomlar için generalize edilmemiştir. Fakat polinomun diskriminantı kavramı yine de kullanışlıdır. <a href="/wiki/Do%C4%9Frusal_cebir" class="mw-redirect" title="Doğrusal cebir">Doğrusal cebir</a> içinde bir endomorfizim minimal polinomunda çoklu köklerin mevcut bulunması endomorfizmin tabiatını değiştirir. Bu şekilde mevcudiyet <a href="/w/index.php?title=Diagonalle%C5%9Ftirme&action=edit&redlink=1" class="new" title="Diagonalleştirme (sayfa mevcut değil)">diagonalleştirme</a> operasyonu imkânsiz yapar. Bu açıklama rasyonel sayıları da içine aldığında, indirgenemeyen polinomların (yani faktorize edilemeyenler) çoklu köklerinin bulunmasi her türlü hal için imkânsızdır. Bu hal tüm haller için gerçek değildir. <a href="/wiki/Galois_teorisi" title="Galois teorisi">Galois teorisi</a> içinde yapılan bu ayrım önemlidir ve sonuçlar konfigürasyona bağlı olarak değişik olabilir. </p><p><br /> </p> <div class="mw-heading mw-heading3"><h3 id="Örnekler_2"><span id=".C3.96rnekler_2"></span>Örnekler</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diskriminant&veaction=edit&section=8" title="Değiştirilen bölüm: Örnekler" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Diskriminant&action=edit&section=8" title="Bölümün kaynak kodunu değiştir: Örnekler"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>İkinci derece polinomlar için ve matris notasyonu kullanarak şu ifade ele geçirilir :</li></ul> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (P)={\frac {(-1)^{\frac {2(2-1)}{2}}}{a}}{\begin{vmatrix}&a&2a&0&\\&b&b&2a&\\&c&0&b&\\\end{vmatrix}}=-{\begin{vmatrix}&1&2&0&\\&b&b&2a&\\&c&0&b&\\\end{vmatrix}}=-{\begin{vmatrix}b&2a&\\0&b&\\\end{vmatrix}}+2{\begin{vmatrix}b&2a&\\c&b&\\\end{vmatrix}}=b^{2}-4ac}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> <mi>a</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd /> <mtd> <mi>a</mi> </mtd> <mtd> <mn>2</mn> <mi>a</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mi>b</mi> </mtd> <mtd> <mi>b</mi> </mtd> <mtd> <mn>2</mn> <mi>a</mi> </mtd> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mi>c</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>b</mi> </mtd> <mtd /> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd /> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mi>b</mi> </mtd> <mtd> <mi>b</mi> </mtd> <mtd> <mn>2</mn> <mi>a</mi> </mtd> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mi>c</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>b</mi> </mtd> <mtd /> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mn>2</mn> <mi>a</mi> </mtd> <mtd /> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>b</mi> </mtd> <mtd /> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mn>2</mn> <mi>a</mi> </mtd> <mtd /> </mtr> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mi>b</mi> </mtd> <mtd /> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>a</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta (P)={\frac {(-1)^{\frac {2(2-1)}{2}}}{a}}{\begin{vmatrix}&a&2a&0&\\&b&b&2a&\\&c&0&b&\\\end{vmatrix}}=-{\begin{vmatrix}&1&2&0&\\&b&b&2a&\\&c&0&b&\\\end{vmatrix}}=-{\begin{vmatrix}b&2a&\\0&b&\\\end{vmatrix}}+2{\begin{vmatrix}b&2a&\\c&b&\\\end{vmatrix}}=b^{2}-4ac}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc06f78afd76ebdbec3063cebb40a016f39acec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:98.295ex; height:9.509ex;" alt="{\displaystyle \Delta (P)={\frac {(-1)^{\frac {2(2-1)}{2}}}{a}}{\begin{vmatrix}&a&2a&0&\\&b&b&2a&\\&c&0&b&\\\end{vmatrix}}=-{\begin{vmatrix}&1&2&0&\\&b&b&2a&\\&c&0&b&\\\end{vmatrix}}=-{\begin{vmatrix}b&2a&\\0&b&\\\end{vmatrix}}+2{\begin{vmatrix}b&2a&\\c&b&\\\end{vmatrix}}=b^{2}-4ac}"></span></center> <ul><li>Üçüncü derecede polinomları için genellikle normalize edilmiş polinom, yani ana diagonal elemanlarının hepsi 1'e eşit olan matrix, kullanılır ve şu ifade ortaya çıkar:</li></ul> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=X^{3}+aX^{2}+bX+c\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>X</mi> <mo>+</mo> <mi>c</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=X^{3}+aX^{2}+bX+c\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed03d4874638e5619fde4e21153efa971a39be1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:25.327ex; height:2.843ex;" alt="{\displaystyle P=X^{3}+aX^{2}+bX+c\;}"></span></center> <p>Bundan şu formül çıkartılır:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (P)=(-1)^{\frac {3(3-1)}{2}}{\begin{vmatrix}&1&a&b&c&0\\&0&1&a&b&c&\\&3&2b&c&0&0&\\&0&3&2b&c&0&\\&0&0&3&2b&c&\\\end{vmatrix}}=a^{2}b^{2}+18abc-4b^{3}-4a^{3}c-27c^{2}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mn>3</mn> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd /> <mtd> <mn>1</mn> </mtd> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> <mtd> <mi>c</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> <mtd> <mi>c</mi> </mtd> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mn>3</mn> </mtd> <mtd> <mn>2</mn> <mi>b</mi> </mtd> <mtd> <mi>c</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>2</mn> <mi>b</mi> </mtd> <mtd> <mi>c</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>2</mn> <mi>b</mi> </mtd> <mtd> <mi>c</mi> </mtd> <mtd /> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>18</mn> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>−</mo> <mn>4</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>c</mi> <mo>−</mo> <mn>27</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta (P)=(-1)^{\frac {3(3-1)}{2}}{\begin{vmatrix}&1&a&b&c&0\\&0&1&a&b&c&\\&3&2b&c&0&0&\\&0&3&2b&c&0&\\&0&0&3&2b&c&\\\end{vmatrix}}=a^{2}b^{2}+18abc-4b^{3}-4a^{3}c-27c^{2}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b92f476e99eac3b1c29bbf8785360b9914be588" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:80.355ex; height:15.843ex;" alt="{\displaystyle \Delta (P)=(-1)^{\frac {3(3-1)}{2}}{\begin{vmatrix}&1&a&b&c&0\\&0&1&a&b&c&\\&3&2b&c&0&0&\\&0&3&2b&c&0&\\&0&0&3&2b&c&\\\end{vmatrix}}=a^{2}b^{2}+18abc-4b^{3}-4a^{3}c-27c^{2}\;}"></span></center> <p>Bu ifade epey karmaşık görünmektedir; fakat bunun bir uygun nedeni vardır. Geleneksel olarak bu karmaşık ifade kullanılırsa yapılan ikamelerle şu şeklide bir polinom elde edilebilir ve bunun diskriminantı gayet basittir: </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=X^{3}+pX+q\quad {\text{et}}\quad \Delta (P)=-4p^{3}-27q^{2}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>p</mi> <mi>X</mi> <mo>+</mo> <mi>q</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>et</mtext> </mrow> <mspace width="1em" /> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>4</mn> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>27</mn> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=X^{3}+pX+q\quad {\text{et}}\quad \Delta (P)=-4p^{3}-27q^{2}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c70c70bd362b46996f14fa4114506ed92cf51d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.104ex; height:3.176ex;" alt="{\displaystyle P=X^{3}+pX+q\quad {\text{et}}\quad \Delta (P)=-4p^{3}-27q^{2}\;}"></span></center> <p>Gerçel katsayılı 3.derece polinom denklemi halinde, eğer diskriminant kesinlikle negatif ise denklemin üç tane ayrı değerde gerçel çözümü bulunur; eğer determinant sıfır ise üç tane birbirine çakışan tek bir gerçel değerde çözüm vardır ve eğer determinant kesinlikle pozitif ise tek bir gerçel çözüm bulunup diğer iki tane çözüm ise birbirlerine <i>conjuge</i> kompleks sayılardır. </p> <ul><li><a href="/w/index.php?title=Elips_e%C4%9Frileri&action=edit&redlink=1" class="new" title="Elips eğrileri (sayfa mevcut değil)">Eliptik Eğriler</a> iki değişkenli üçüncü derece polinomların özel bir şeklinden ortaya çıkarlar.</li></ul> <p>Eliptik eğrinin en basit bir halinde (kısa Weierstrass Eliptik Eğrisi) denklem şöyledir: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}=x^{3}+ax+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=x^{3}+ax+b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dbe6cab1bc2c7f1c99757dc6e5d7a517cf9b4f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.935ex; height:3.009ex;" alt="{\displaystyle y^{2}=x^{3}+ax+b}"></span> Bunda <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181523deba732fda302fd176275a0739121d3bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}"></span> katsayıları gerçel sayılardır. Bu halde diskriminant şöyle tanımlanır: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =-16(4a^{3}+27b^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mo>−</mo> <mn>16</mn> <mo stretchy="false">(</mo> <mn>4</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>27</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =-16(4a^{3}+27b^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bdec8937e8daec088f44f0302602dcbc4cc0c91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.64ex; height:3.176ex;" alt="{\displaystyle \Delta =-16(4a^{3}+27b^{2})}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Genel_şekilde_ifade"><span id="Genel_.C5.9Fekilde_ifade"></span>Genel şekilde ifade</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diskriminant&veaction=edit&section=9" title="Değiştirilen bölüm: Genel şekilde ifade" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Diskriminant&action=edit&section=9" title="Bölümün kaynak kodunu değiştir: Genel şekilde ifade"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>P</i> dereceli polinom için genel diskriminant ifadesi şöyle tanımlanır: </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots +a_{1}X+a_{0}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>X</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots +a_{1}X+a_{0}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62fd306146827cea335f08723112c846fce8d13a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:41.65ex; height:3.009ex;" alt="{\displaystyle P=a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots +a_{1}X+a_{0}\;}"></span></center> <p>ve bundan şu ortaya çıkar: </p> <div><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (P)=(-1)^{\frac {n(n-1)}{2}}{\begin{vmatrix}1&0&\cdots &0&n&0&\cdots &0\\a_{n-1}&a_{n}&\ddots &\vdots &(n-1)a_{n-1}&na_{n}&\ddots &\vdots \\\vdots &a_{n-1}&\ddots &0&\vdots &(n-1)a_{n-1}&\ddots &0\\a_{0}&\vdots &\ddots &a_{n}&a_{0}&\vdots &\ddots &na_{n}\\0&a_{0}&&a_{n-1}&0&a_{0}&&(n-1)a_{n-1}\\\vdots &\ddots &\ddots &\vdots &\vdots &\ddots &\ddots &\vdots \\0&\cdots &0&a_{0}&0&\cdots &0&a_{0}\\\end{vmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>n</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi>n</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mi>n</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd /> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd /> <mtd> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta (P)=(-1)^{\frac {n(n-1)}{2}}{\begin{vmatrix}1&0&\cdots &0&n&0&\cdots &0\\a_{n-1}&a_{n}&\ddots &\vdots &(n-1)a_{n-1}&na_{n}&\ddots &\vdots \\\vdots &a_{n-1}&\ddots &0&\vdots &(n-1)a_{n-1}&\ddots &0\\a_{0}&\vdots &\ddots &a_{n}&a_{0}&\vdots &\ddots &na_{n}\\0&a_{0}&&a_{n-1}&0&a_{0}&&(n-1)a_{n-1}\\\vdots &\ddots &\ddots &\vdots &\vdots &\ddots &\ddots &\vdots \\0&\cdots &0&a_{0}&0&\cdots &0&a_{0}\\\end{vmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb4da7610cf7df73195d00e36b4d0e02de877e15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.171ex; width:91.957ex; height:29.509ex;" alt="{\displaystyle \Delta (P)=(-1)^{\frac {n(n-1)}{2}}{\begin{vmatrix}1&0&\cdots &0&n&0&\cdots &0\\a_{n-1}&a_{n}&\ddots &\vdots &(n-1)a_{n-1}&na_{n}&\ddots &\vdots \\\vdots &a_{n-1}&\ddots &0&\vdots &(n-1)a_{n-1}&\ddots &0\\a_{0}&\vdots &\ddots &a_{n}&a_{0}&\vdots &\ddots &na_{n}\\0&a_{0}&&a_{n-1}&0&a_{0}&&(n-1)a_{n-1}\\\vdots &\ddots &\ddots &\vdots &\vdots &\ddots &\ddots &\vdots \\0&\cdots &0&a_{0}&0&\cdots &0&a_{0}\\\end{vmatrix}}}"></span></div> <div class="mw-heading mw-heading2"><h2 id="Diskriminant_cebirsel_tam_sayılar_halkası"><span id="Diskriminant_cebirsel_tam_say.C4.B1lar_halkas.C4.B1"></span>Diskriminant cebirsel tam sayılar halkası</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diskriminant&veaction=edit&section=10" title="Değiştirilen bölüm: Diskriminant cebirsel tam sayılar halkası" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Diskriminant&action=edit&section=10" title="Bölümün kaynak kodunu değiştir: Diskriminant cebirsel tam sayılar halkası"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sayilar cebiri teorisi tanımı farklı görünen bir diskriminant kavramı kullanır.Bu kavram bir kuadratik formdaki determinanta karşıttır ve matamati <i>halka</i> <i>A</i> için kullanılır. Her diskriminantın her iki tanımı da birbiriyle çok yakın olarak bağlıdırlar. </p><p>Eğer <i>A</i> halkasını(tümüyle relatiflerden oluşan bir <i>Z</i> için) <i>Z</i>[<i>a</i>] ile eşit yapan bir <a href="/w/index.php?title=Cebirsel_tam_say%C4%B1&action=edit&redlink=1" class="new" title="Cebirsel tam sayı (sayfa mevcut değil)">cebirsel tam sayı</a> <i>a</i> mevcutsa, <i>a</i> için <a href="/w/index.php?title=Minimal_polinom&action=edit&redlink=1" class="new" title="Minimal polinom (sayfa mevcut değil)">minimal polinom</a> <i>Z</i> içindeki katsayılari aynen içerir <i>A'</i>nın polinomlara gore tanımlanmış anlamı ile cebirsel sayı teorisine göre <i>halka</i>nın diskriminantı anlamı ile tamamına eşittir. </p> <div class="mw-heading mw-heading2"><h2 id="Kaynakça"><span id="Kaynak.C3.A7a"></span>Kaynakça</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diskriminant&veaction=edit&section=11" title="Değiştirilen bölüm: Kaynakça" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Diskriminant&action=edit&section=11" title="Bölümün kaynak kodunu değiştir: Kaynakça"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r32805677">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-count:2}.mw-parser-output .reflist-columns-3{column-count:3}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><strong><a href="#cite_ref-1">^</a></strong> <span class="reference-text">Örneğin bu formül "Encyclopédia Britanıca" "discriminant" maddesinde bulunur <a rel="nofollow" class="external autonumber" href="http://www.britannica.com/eb/artıcle-9030624/discriminant">[1]</a></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Dış_bağlantılar"><span id="D.C4.B1.C5.9F_ba.C4.9Flant.C4.B1lar"></span>Dış bağlantılar</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diskriminant&veaction=edit&section=12" title="Değiştirilen bölüm: Dış bağlantılar" class="mw-editsection-visualeditor"><span>değiştir</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Diskriminant&action=edit&section=12" title="Bölümün kaynak kodunu değiştir: Dış bağlantılar"><span>kaynağı değiştir</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>E. W. Weisstein, "Polinom dikriminant" <i>Wolfram MathWorld</i> <a rel="nofollow" class="external autonumber" href="http://mathworld.wolfram.com/PolynomialDiscriminant.html">[2]</a>15 Eylül 2008 tarihinde <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> sitesinde <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080915084049/http://mathworld.wolfram.com/PolynomialDiscriminant.html">arşivlendi</a>. <span style="font-size:0.95em; font-weight:bold; color:inherit;">(İngilizce)</span> (Erişim:12.1.2010)</li> <li>W.D.Nickalls ve R.H.D Rye (1996 Temmuz) "Bir polinomin diskriminantinin geometrisi" <i>The Mathematical Gazette'</i> Cilt: 80 Sayfa:279–285: <a rel="nofollow" class="external autonumber" href="https://web.archive.org/web/20080725082359/http://www.m-a.org.uk/docs/library/2548.pdf">[3]</a> <span style="font-size:0.95em; font-weight:bold; color:inherit;">(İngilizce)</span> (Erişim:12.1.2010)</li></ul></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">"<a dir="ltr" href="https://tr.wikipedia.org/w/index.php?title=Diskriminant&oldid=32761324">https://tr.wikipedia.org/w/index.php?title=Diskriminant&oldid=32761324</a>" sayfasından alınmıştır</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%C3%96zel:Kategoriler" title="Özel:Kategoriler">Kategori</a>: <ul><li><a href="/wiki/Kategori:Determinant" title="Kategori:Determinant">Determinant</a></li><li><a href="/wiki/Kategori:Cebirsel_say%C4%B1_teorisi" title="Kategori:Cebirsel sayı teorisi">Cebirsel sayı teorisi</a></li><li><a href="/wiki/Kategori:Konik_kesitler" title="Kategori:Konik kesitler">Konik kesitler</a></li><li><a href="/wiki/Kategori:Polinomlar" title="Kategori:Polinomlar">Polinomlar</a></li><li><a href="/wiki/Kategori:Kuadratik_formlar" title="Kategori:Kuadratik formlar">Kuadratik formlar</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Gizli kategoriler: <ul><li><a href="/wiki/Kategori:K%C4%B1rm%C4%B1z%C4%B1_ba%C4%9Flant%C4%B1ya_sahip_ana_madde_%C5%9Fablonu_i%C3%A7eren_maddeler" title="Kategori:Kırmızı bağlantıya sahip ana madde şablonu içeren maddeler">Kırmızı bağlantıya sahip ana madde şablonu içeren maddeler</a></li><li><a href="/wiki/Kategori:Webar%C5%9Fiv_%C5%9Fablonu_wayback_ba%C4%9Flant%C4%B1lar%C4%B1" title="Kategori:Webarşiv şablonu wayback bağlantıları">Webarşiv şablonu wayback bağlantıları</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Sayfa en son 14.01, 12 Mayıs 2024 tarihinde değiştirildi.</li> <li id="footer-info-copyright">Metin <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/deed.tr">Creative Commons Atıf-AynıLisanslaPaylaş Lisansı</a> altındadır ve ek koşullar uygulanabilir. 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