Szerkesztő:Gubbubu – Wikikönyvek

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class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Bevezető</div> </a> </li> <li id="toc-Aktuális" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Aktuális"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Aktuális</span> </div> </a> <button aria-controls="toc-Aktuális-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>A(z) Aktuális alszakasz kinyitása/becsukása</span> </button> <ul id="toc-Aktuális-sublist" class="vector-toc-list"> <li id="toc-Adminisztrálás" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Adminisztrálás"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Adminisztrálás</span> </div> </a> <ul id="toc-Adminisztrálás-sublist" 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aria-controls="toc-Bizonyítások-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>A(z) Bizonyítások alszakasz kinyitása/becsukása</span> </button> <ul id="toc-Bizonyítások-sublist" class="vector-toc-list"> <li id="toc-Elemi_algebrai_eszközökre_épülő_bizonyítás" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elemi_algebrai_eszközökre_épülő_bizonyítás"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Elemi algebrai eszközökre épülő bizonyítás</span> </div> </a> <ul id="toc-Elemi_algebrai_eszközökre_épülő_bizonyítás-sublist" class="vector-toc-list"> </ul> </li> </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="Tartalomjegyzék" 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="Tartalomjegyzék kinyitása/becsukása" > <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">Tartalomjegyzék kinyitása/becsukása</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-namespace">Szerkesztő</span><span class="mw-page-title-separator">:</span><span class="mw-page-title-main">Gubbubu</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="Ugrás egy más nyelvű szócikkre. Elérhető 1 nyelven" > <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-1" 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">1 nyelv</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikibooks.org/wiki/User:Gubbubu" title="User:Gubbubu – angol" lang="en" hreflang="en" data-title="User:Gubbubu" data-language-autonym="English" data-language-local-name="angol" class="interlanguage-link-target"><span>English</span></a></li> </ul> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Névterek"> <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-user" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Szerkeszt%C5%91:Gubbubu" title="A felhasználói lap megtekintése [c]" accesskey="c"><span>Userlap</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Szerkeszt%C5%91vita:Gubbubu" rel="discussion" title="Az oldal tartalmának megvitatása [t]" accesskey="t"><span>Vitalap</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="Nyelvvariáns váltása" > <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">magyar</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="Nézetek"> <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/Szerkeszt%C5%91:Gubbubu"><span>Olvasás</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Szerkeszt%C5%91:Gubbubu&action=edit" title="A lap szerkesztése [e]" accesskey="e"><span>Szerkesztés</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Szerkeszt%C5%91:Gubbubu&action=history" title="A lap korábbi változatai [h]" accesskey="h"><span>Laptörténet</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Oldal eszközök"> <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="Eszközök" > <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">Eszközök</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">Eszközök</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">áthelyezés az oldalsávba</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">elrejtés</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="További lehetőségek" > <div class="vector-menu-heading"> Műveletek </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/Szerkeszt%C5%91:Gubbubu"><span>Olvasás</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Szerkeszt%C5%91:Gubbubu&action=edit" title="A lap szerkesztése [e]" accesskey="e"><span>Szerkesztés</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Szerkeszt%C5%91:Gubbubu&action=history"><span>Laptörténet</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Általános </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Mi_hivatkozik_erre/Szerkeszt%C5%91:Gubbubu" title="Az erre a lapra hivatkozó más lapok listája [j]" accesskey="j"><span>Mi hivatkozik erre?</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Kapcsol%C3%B3d%C3%B3_v%C3%A1ltoztat%C3%A1sok/Szerkeszt%C5%91:Gubbubu" rel="nofollow" title="Az erről a lapról hivatkozott lapok utolsó változtatásai [k]" accesskey="k"><span>Kapcsolódó változtatások</span></a></li><li id="t-contributions" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Szerkeszt%C5%91_k%C3%B6zrem%C5%B1k%C3%B6d%C3%A9sei/Gubbubu" title="A felhasználó közreműködéseinek listája"><span>Felhasználó közreműködései</span></a></li><li id="t-log" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Rendszernapl%C3%B3k/Gubbubu"><span>Rendszernaplók</span></a></li><li id="t-userrights" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Szerkeszt%C5%91k_jogai/Gubbubu"><span>Felhasználói csoportok megtekintése</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Speci%C3%A1lis_lapok" title="Az összes speciális lap listája [q]" accesskey="q"><span>Speciális lapok</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Szerkeszt%C5%91:Gubbubu&oldid=351931" title="Állandó hivatkozás ezen lap ezen változatához"><span>Hivatkozás erre a változatra</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Szerkeszt%C5%91:Gubbubu&action=info" title="További információk erről a lapról"><span>Lapinformációk</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:UrlShortener&url=https%3A%2F%2Fhu.wikibooks.org%2Fwiki%2FSzerkeszt%25C5%2591%3AGubbubu"><span>Rövidített URL készítése</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:QrCode&url=https%3A%2F%2Fhu.wikibooks.org%2Fwiki%2FSzerkeszt%25C5%2591%3AGubbubu"><span>QR-kód letöltése</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"> Nyomtatás/exportálás </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=Speci%C3%A1lis:K%C3%B6nyv&bookcmd=book_creator&referer=Szerkeszt%C5%91%3AGubbubu"><span>Könyv készítése</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:DownloadAsPdf&page=Szerkeszt%C5%91%3AGubbubu&action=show-download-screen"><span>Letöltés PDF-ként</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Szerkeszt%C5%91:Gubbubu&printable=yes" title="A lap nyomtatható változata [p]" accesskey="p"><span>Nyomtatható változat</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects emptyPortlet" > <div class="vector-menu-heading"> Társprojektek </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </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="Oldal eszközök"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Megjelenés"> <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">Megjelenés</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">áthelyezés az oldalsávba</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">elrejtés</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">A Wikikönyvekből, a szabad elektronikus könyvtárból.</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="hu" dir="ltr"><p><a class="external text" href="https://en.wikipedia.org/wiki/User:Gubbubu">Lásd itt</a> </p><p><br /> </p> <table class="babel_box mw-babel-wrapper" style="float: right; margin-left: 1em; margin-bottom: 0.5em; width: 242px; border: #99B3FF solid 1px"> <tbody><tr> <th class="mw-babel-header"><a href="/wiki/Seg%C3%ADts%C3%A9g:B%C3%A1bel" title="Segítség:Bábel">Bábel szerkesztői információk</a> </th></tr> <tr> <td><div class="mw-babel-box mw-babel-box-N" style="float: left; border: 1px solid #6EF7A7; margin: 1px"> <table cellspacing="0" style="width: 238px; background-color: #C5FCDC"> <tbody><tr> <th style="width: 45px; height: 45px; background-color: #6EF7A7; text-align: center; font-size: 14pt">hu</th> <td style="font-size: 8pt; padding: 4pt; line-height: 1.25em; text-align: center;">Ennek a szerkesztőnek <a href="/wiki/Kateg%C3%B3ria:User_hu" title="Kategória:User hu">magyar</a> az <a href="/wiki/Kateg%C3%B3ria:User_hu-N" title="Kategória:User hu-N">anyanyelve</a>.</td> </tr> </tbody></table> </div> </td></tr> <tr> <td><div class="mw-babel-box mw-babel-box-3" style="float: left; border: 1px solid #99B3FF; margin: 1px"> <table cellspacing="0" style="width: 238px; background-color: #E0E8FF"> <tbody><tr> <th style="width: 45px; height: 45px; background-color: #99B3FF; text-align: center; font-size: 14pt">en-3</th> <td style="font-size: 8pt; padding: 4pt; line-height: 1.25em; text-align: center;">This user has <a href="/wiki/Kateg%C3%B3ria:User_en-3" title="Kategória:User en-3">advanced</a> knowledge of <a href="/wiki/Kateg%C3%B3ria:User_en" title="Kategória:User en">English</a>.</td> </tr> </tbody></table> </div> </td></tr> <tr> <td><div class="mw-babel-box mw-babel-box-1" style="float: left; border: 1px solid #C0C8FF; margin: 1px"> <table cellspacing="0" style="width: 238px; background-color: #F0F8FF"> <tbody><tr> <th style="width: 45px; height: 45px; background-color: #C0C8FF; text-align: center; font-size: 14pt">la-1</th> <td style="font-size: 8pt; padding: 4pt; line-height: 1.25em; text-align: center;">Hic usor/Haec ustrix <a href="/w/index.php?title=Kateg%C3%B3ria:User_la-1&action=edit&redlink=1" class="new" title="Kategória:User la-1 (a lap nem létezik)">simplici</a> <a href="/wiki/Kateg%C3%B3ria:User_la" title="Kategória:User la">lingua Latina</a> conferre potest.</td> </tr> </tbody></table> </div> </td></tr> <tr> <th class="mw-babel-footer"><a href="/wiki/Kateg%C3%B3ria:Felhaszn%C3%A1l%C3%B3k_nyelvei" title="Kategória:Felhasználók nyelvei">Felhasználók keresése nyelv szerint</a> </th></tr></tbody></table> <table> <tbody><tr> <td><div style="float: left; border: solid #77E0E8 1px; margin: 1px;"> <table cellspacing="0" style="width: 238px; background: #D0F8FF;"> <tbody><tr> <td style="width: 45px; height: 45px; background: #77E0E8; text-align: center; font-size: 10pt;"><b>html-2</b> </td> <td style="font-size: 8pt; padding: 4pt; line-height: 1.25em;">Ez a felhasználó <b><a href="/w/index.php?title=Kateg%C3%B3ria:User_html-2&action=edit&redlink=1" class="new" title="Kategória:User html-2 (a lap nem létezik)">közepes</a> szintű</b> <a href="/wiki/HTML" class="mw-redirect" title="HTML">HTML</a> tudással rendelkezik. </td></tr></tbody></table></div> </td></tr></tbody></table> <p><br /> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Aktuális"><span id="Aktu.C3.A1lis"></span>Aktuális</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Szerkeszt%C5%91:Gubbubu&action=edit&section=1" title="Szakasz szerkesztése: Aktuális"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Szerkeszt%C5%91:Gubbubu/Linkgy%C5%B1jtem%C3%A9ny" title="Szerkesztő:Gubbubu/Linkgyűjtemény">Szerkesztő:Gubbubu/Linkgyűjtemény</a> (Hasznos, többnyire ismeretterjesztő linkek különféle érdeklődési területeimről a matematikafilozófiától kezdve a popzenetörténeten és klasszikafilológián át egészen az absztrakt festészet elméletéig).</li> <li><a href="/wiki/Szerkeszt%C5%91:Gubbubu/Linkgy%C5%B1jtem%C3%A9ny/Szem%C3%A9lyes" title="Szerkesztő:Gubbubu/Linkgyűjtemény/Személyes">Szerkesztő:Gubbubu/Linkgyűjtemény/Személyes</a> (kedvenc vagy utált zenék, kedvenc filmek, kedvenc szintik stb.)</li> <li><a href="/wiki/Szerkeszt%C5%91:Gubbubu/Audiocuccol%C3%B3gia" title="Szerkesztő:Gubbubu/Audiocuccológia">Szerkesztő:Gubbubu/Audiocuccológia</a> (szintetizátorok és hasonló audioeszközök, amiket életem során huzamosabb ideig birtokoltam)</li></ul> <ul><li>Linkográfiák (popzene, elsősorban):</li> <li><a href="/w/index.php?title=Szerkeszt%C5%91:Gubbubu/Alice_Cooper_linkogr%C3%A1fia&action=edit&redlink=1" class="new" title="Szerkesztő:Gubbubu/Alice Cooper linkográfia (a lap nem létezik)">Szerkesztő:Gubbubu/Alice Cooper linkográfia</a></li> <li><a href="/w/index.php?title=Szerkeszt%C5%91:Gubbubu/Depeche_Mode_linkogr%C3%A1fia&action=edit&redlink=1" class="new" title="Szerkesztő:Gubbubu/Depeche Mode linkográfia (a lap nem létezik)">SzerkesztŐ:Gubbubu/Depeche Mode linkográfia</a></li> <li><a href="/wiki/Szerkeszt%C5%91:Gubbubu/Fluke_linkogr%C3%A1fia" title="Szerkesztő:Gubbubu/Fluke linkográfia">Szerkesztő:Gubbubu/Fluke linkográfia</a></li> <li><a href="/wiki/Szerkeszt%C5%91:Gubbubu/Icehouse_linkogr%C3%A1fia" title="Szerkesztő:Gubbubu/Icehouse linkográfia">Szerkesztő:Gubbubu/Icehouse linkográfia</a></li> <li><a href="/w/index.php?title=Szerkeszt%C5%91:Gubbubu/Tangerine_Dream_linkogr%C3%A1fia&action=edit&redlink=1" class="new" title="Szerkesztő:Gubbubu/Tangerine Dream linkográfia (a lap nem létezik)">Szerkesztő:Gubbubu/Tangerine Dream linkográfia</a></li> <li><a href="/w/index.php?title=Szerkeszt%C5%91:Gubbubu/Yello_linkogr%C3%A1fia&action=edit&redlink=1" class="new" title="Szerkesztő:Gubbubu/Yello linkográfia (a lap nem létezik)">Szerkesztő:Gubbubu/Yello linkográfia</a></li></ul> <p>Ezek a lapok eredetileg arra szolgáltak, hogy zenét hallgathassak wikimunka közben. A <a href="https://hu.wikipedia.org/wiki/hu:Deezer" class="extiw" title="w:hu:Deezer">Deezer</a> és <a href="https://hu.wikipedia.org/wiki/hu:Spotify" class="extiw" title="w:hu:Spotify">Spotify</a> stb. megjelenése, valamint a Youtube lassú jogászi kinyírása és állandó jogászi úton való zaklatása (videók eltűnnek, törlődnek, beprivátulnak) óta ez a funkció jelentőségében csökkent. De ha valahol pl. jó minőségben fellelhető egy videoklip, az mindig fontos info és segít a zenekar cikkének megírásában is. </p> <ul><li><a href="/wiki/Szerkeszt%C5%91:Gubbubu/MatTanterv" title="Szerkesztő:Gubbubu/MatTanterv">Szerkesztő:Gubbubu/MatTanterv</a></li> <li><a href="/wiki/Szerkeszt%C5%91:Gubbubu/Elemi_logika_feladatgy%C5%B1jtem%C3%A9ny" title="Szerkesztő:Gubbubu/Elemi logika feladatgyűjtemény">Szerkesztő:Gubbubu/Elemi logika feladatgyűjtemény</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Adminisztrálás"><span id="Adminisztr.C3.A1l.C3.A1s"></span>Adminisztrálás</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Szerkeszt%C5%91:Gubbubu&action=edit&section=2" title="Szakasz szerkesztése: Adminisztrálás"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Wikik%C3%B6nyvek:Friss_esem%C3%A9nyek" title="Wikikönyvek:Friss események">Wikikönyvek / Friss Események</a></li></ul> <p>"Valaki lebutította a rendszert,törölték a segédeszközöket (pl.: látszer) ezért nehezebbé vált az ellenőrzés és a szerkesztés is. A szerkesztőlap tetején lévő gombok egy részét törölték! Kinek volt útjába? A "Friss Változtatás"okban nem mutatja az eltéréseket, így nehezebb ellenőrizni a vandál bejegyzéseket is. Nincs gazdája (adminja) a projektnek, mert csak nagyon ritkán jelenik meg az 5 admin közül egy (általában Gubbubu)" </p> <ul><li><a href="/wiki/Kateg%C3%B3ria:Azonnali_t%C3%B6rl%C3%A9sre_v%C3%A1r%C3%B3_lapok" title="Kategória:Azonnali törlésre váró lapok">Kategória:Azonnali törlésre váró lapok</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Régi"><span id="R.C3.A9gi"></span>Régi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Szerkeszt%C5%91:Gubbubu&action=edit&section=3" title="Szakasz szerkesztése: Régi"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Szépen halad: </p> <ul><li><a href="/w/index.php?title=Halmazelm%C3%A9let&action=edit&redlink=1" class="new" title="Halmazelmélet (a lap nem létezik)">Halmazelmélet</a></li></ul> <p>előkészületben: </p> <ul><li><a href="/wiki/Szerkeszt%C5%91:Gubbubu/J%C3%A1t%C3%A9kosszint%C5%B1_AI-szkriptek_%C3%ADr%C3%A1sa_a_Star_Wars_Galactic_Battlegrounds_c._val%C3%B3s_idej%C5%B1_strat%C3%A9giai_videoj%C3%A1t%C3%A9khoz" class="mw-redirect" title="Szerkesztő:Gubbubu/Játékosszintű AI-szkriptek írása a Star Wars Galactic Battlegrounds c. valós idejű stratégiai videojátékhoz">Játékosszintű AI-szkriptek írása a Star Wars Galactic Battlegrounds c. valós idejű stratégiai videojátékhoz</a></li> <li><a href="/wiki/Line%C3%A1ris_algebra" title="Lineáris algebra">Lineáris algebra</a></li> <li><a href="/w/index.php?title=Csoportelm%C3%A9let&action=edit&redlink=1" class="new" title="Csoportelmélet (a lap nem létezik)">Csoportelmélet</a></li> <li><a href="/wiki/Sz%C3%A1melm%C3%A9let" title="Számelmélet">Számelmélet</a></li> <li><a href="/w/index.php?title=Differenci%C3%A1legyenletek&action=edit&redlink=1" class="new" title="Differenciálegyenletek (a lap nem létezik)">Differenciálegyenletek</a> - néhány alapdolog a differenciálegyenletekről, 20-30 oldalas régi általam írt jegyzetet kell átwikisíteni, semmi komolytól nem kell tartani tehát (komplex analízistól, meg ilyesmiktől).</li> <li><a href="/w/index.php?title=Latin_nyelvtan&action=edit&redlink=1" class="new" title="Latin nyelvtan (a lap nem létezik)">Latin nyelvtan (alapfok)</a> ha esetleg lesz időm.</li> <li><a href="/wiki/Szerkeszt%C5%91:Gubbubu/Halmazrendszerek_geometri%C3%A1ja" title="Szerkesztő:Gubbubu/Halmazrendszerek geometriája">Halmazrendszerek geometriája</a></li> <li><a href="/wiki/Szerkeszt%C5%91:Gubbubu/monobook.css" title="Szerkesztő:Gubbubu/monobook.css">User:Gubbubu/monobook.css</a></li> <li><a href="/wiki/Szerkeszt%C5%91:Gubbubu/Arpadgabor" title="Szerkesztő:Gubbubu/Arpadgabor">User:Gubbubu/Arpadgabor</a></li> <li><a href="/wiki/Szerkeszt%C5%91:Gubbubu/K%C3%B6nyvadatb%C3%A1zis" title="Szerkesztő:Gubbubu/Könyvadatbázis">User:Gubbubu/Könyvadatbázis</a></li></ul> <ul><li><a href="/wiki/Szerkeszt%C5%91:Gubbubu/Miscmatek" title="Szerkesztő:Gubbubu/Miscmatek">User:Gubbubu/Miscmatek</a></li> <li><a href="/wiki/Szerkeszt%C5%91:Gubbubu/Elemi_geometria" title="Szerkesztő:Gubbubu/Elemi geometria">User:Gubbubu/Elemi geometria</a></li> <li><a href="/wiki/Szerkeszt%C5%91:Gubbubu/Matematikai_logika" title="Szerkesztő:Gubbubu/Matematikai logika">User:Gubbubu/Matematikai logika</a></li> <li><a href="/wiki/Szerkeszt%C5%91:Gubbubu/Alairas" title="Szerkesztő:Gubbubu/Alairas">Szerkesztő:Gubbubu/Alairas</a></li> <li><a href="/wiki/Szerkeszt%C5%91:KeFe/Vand%C3%A1l" title="Szerkesztő:KeFe/Vandál">User:KeFe/Vandál</a> hasznos segédeszköz</li></ul> <p>A használata pedig: {{fejléc|tartalom=[[Lineáris algebra|Tartalomjegyzék]]|előző=[[Lineáris algebra - 1.|(Bevezetés)]]|következő=[[Lineáris algebra - 3.|(Lineáris egyenletrendszer ekvivalens átalakításai)]]}} </p> <ul><li><a href="/wiki/Sablon:Fv" title="Sablon:Fv">Sablon:Fv</a></li></ul> <p>Mi legyen a kimithisz cikksorozat címe? </p> <ul><li><a href="/w/index.php?title=Bevezet%C3%A9s_a_vall%C3%A1sok_tanaiba&action=edit&redlink=1" class="new" title="Bevezetés a vallások tanaiba (a lap nem létezik)">Bevezetés a vallások tanaiba</a></li> <li><a href="/w/index.php?title=Ki_mit_hisz%3F_-_bevezet%C3%A9s_az_%C3%B6sszehasonl%C3%ADt%C3%B3_vall%C3%A1studom%C3%A1nyba&action=edit&redlink=1" class="new" title="Ki mit hisz? - bevezetés az összehasonlító vallástudományba (a lap nem létezik)">Ki mit hisz? - bevezetés az összehasonlító vallástudományba</a></li> <li>vagy csak egyszerűen <a href="/w/index.php?title=Ki_mit_hisz%3F&action=edit&redlink=1" class="new" title="Ki mit hisz? (a lap nem létezik)">Ki mit hisz?</a></li></ul> <p>Még gondolkodok ... <a class="mw-selflink selflink">Gubbubu</a> 2005. július 11., 08:08 (UTC) </p> <div class="mw-heading mw-heading2"><h2 id="Külső_hivatkozások"><span id="K.C3.BCls.C5.91_hivatkoz.C3.A1sok"></span>Külső hivatkozások</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Szerkeszt%C5%91:Gubbubu&action=edit&section=4" title="Szakasz szerkesztése: Külső hivatkozások"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ϯ ϯ </p> <ul><li><a rel="nofollow" class="external autonumber" href="http://64.233.183.104/search?q=cache:0JsCiZ-BIUIJ:www.konyvtar.elte.hu/szolgaltatasok/e_szolg/dr_temat.htm+wikibooks&hl=hu&lr=lang_hu">[1]</a></li></ul> <hr /> <ul><li><a class="external text" href="https://hu.wikibooks.org/w/index.php?title=Speci%C3%A1lis:Contributions&target=83.216.53.42">83.216.53.42</a></li> <li><a class="external text" href="https://hu.wikibooks.org/wiki/Speci%C3%A1lis:Szerkeszt%C5%91_k%C3%B6zrem%C5%B1k%C3%B6d%C3%A9sei/84.2.74.2">84.2.74.2</a></li> <li><a class="external autonumber" href="https://hu.wikibooks.org/wiki/Speci%C3%A1lis:Szerkeszt%C5%91_k%C3%B6zrem%C5%B1k%C3%B6d%C3%A9sei/84.0.162.187">[2]</a></li> <li><a href="/wiki/Szerkeszt%C5%91:MediaWiki_default" title="Szerkesztő:MediaWiki default">A legproblémásabb szerkesztő</a></li></ul> <table style="margin-left:auto; margin-right:auto; padding:0; text-align:center; border:1px solid #ffc9c9; background-color:#fffff3; font-size:85%;"> <tbody><tr> <td style="vertical-align:middle; padding:.5em;"><b>Ez a Wikikönyvek egyik felhasználói lapja.</b><br />Ha ezt a lapot nem a Wikikönyvekben olvasod, akkor egy tükrözést látsz. Légy tudatában annak, hogy a lap elavult lehet, és hogy ezen felhasználónak valószínűleg nincs kapcsolata a Wikikönyveken kívül semmilyen más, ezt a lapot tartalmazó weboldallal. Az eredeti felhasználói lapot a <a class="external free" href="https://hu.wikibooks.org/wiki/Szerkeszt%C5%91:Gubbubu">https://hu.wikibooks.org/wiki/Szerkeszt%C5%91:Gubbubu</a> címen találod meg. </td> <td style="vertical-align:middle; padding:.5em;"><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:Wikimedia_Foundation_RGB_logo_with_text.svg" class="mw-file-description" title="Wikimédia Alapítvány"><img alt="Wikimédia Alapítvány" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Wikimedia_Foundation_RGB_logo_with_text.svg/60px-Wikimedia_Foundation_RGB_logo_with_text.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Wikimedia_Foundation_RGB_logo_with_text.svg/90px-Wikimedia_Foundation_RGB_logo_with_text.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Wikimedia_Foundation_RGB_logo_with_text.svg/120px-Wikimedia_Foundation_RGB_logo_with_text.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Bizonyítások"><span id="Bizony.C3.ADt.C3.A1sok"></span>Bizonyítások</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Szerkeszt%C5%91:Gubbubu&action=edit&section=5" title="Szakasz szerkesztése: Bizonyítások"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Elemi_algebrai_eszközökre_épülő_bizonyítás"><span id="Elemi_algebrai_eszk.C3.B6z.C3.B6kre_.C3.A9p.C3.BCl.C5.91_bizony.C3.ADt.C3.A1s"></span>Elemi algebrai eszközökre épülő bizonyítás</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Szerkeszt%C5%91:Gubbubu&action=edit&section=6" title="Szakasz szerkesztése: Elemi algebrai eszközökre épülő bizonyítás"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Legyenek a háromszög csúcsai a szokásos módon A,B,C, a szemközti oldalak a,b,c, T a c ponthoz tartozó m<sub>c</sub> := m magasság talppontja! A magasság az ABC háromszöget két részháromszögre bontja, ezek az ATC és BTC derékszögű háromszögek. Legyen az AT távolság AT=x, ekkor TB=AB-AT=c-x. Felírva a két derékszögű háromszögre Pitagorasz tételét, </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 {\overline {AT}}^{2}+{\overline {TC}}^{2}={\overline {AC}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>T</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>T</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AT}}^{2}+{\overline {TC}}^{2}={\overline {AC}}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2cbe753220a0119d1c812fa4b025959825c46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.055ex; height:3.676ex;" alt="{\displaystyle {\overline {AT}}^{2}+{\overline {TC}}^{2}={\overline {AC}}^{2}}"></span></center> <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 {\overline {BT}}^{2}+{\overline {TC}}^{2}={\overline {BC}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>T</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>T</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {BT}}^{2}+{\overline {TC}}^{2}={\overline {BC}}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f73606a5f7ea50fac1a518103c5bf0cb2384f02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.097ex; height:3.676ex;" alt="{\displaystyle {\overline {BT}}^{2}+{\overline {TC}}^{2}={\overline {BC}}^{2}}"></span></center> <p>azaz </p> <dl><dd><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}+m^{2}=b^{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> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+m^{2}=b^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/759eb77ad302a6206c4c7557bd012904aef149b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.469ex; height:2.843ex;" alt="{\displaystyle x^{2}+m^{2}=b^{2}}"></span></dd> <dd><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 (c-x)^{2}+m^{2}=a^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c-x)^{2}+m^{2}=a^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be12bfb18861966b90618d335d1e455aab92c4a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.358ex; height:3.176ex;" alt="{\displaystyle (c-x)^{2}+m^{2}=a^{2}}"></span></dd></dl> <p>Ismertnek tételezve az a,b,c mennyiségeket, a fenti egyenletrendszer egy algebrai, másodfokú kétismeretlenes egyenletrendszer. Ezt a következő módon kényelmesen meg lehet oldani: fejezzük ki az első egyenletből az m<sup>2</sup> mennyiséget, és helyettesítsük be a második egyenletbe: </p> <dl><dd><dl><dd><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 m^{2}=b^{2}-x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m^{2}=b^{2}-x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d34752db640d864d9262048d4fc799c0aa627550" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.469ex; height:2.843ex;" alt="{\displaystyle m^{2}=b^{2}-x^{2}}"></span>     (*)</dd> <dd><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 (c-x)^{2}+\left(b^{2}-x^{2}\right)=a^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c-x)^{2}+\left(b^{2}-x^{2}\right)=a^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00ec52ac96648eafec98b385bb736079445b2e44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.669ex; height:3.343ex;" alt="{\displaystyle (c-x)^{2}+\left(b^{2}-x^{2}\right)=a^{2}}"></span></dd></dl></dd></dl> <p>A második egyenletben alkalmazva a két tag különbségére vonatkozó <a href="/w/index.php?title=Elemi_algebra&action=edit&redlink=1" class="new" title="Elemi algebra (a lap nem létezik)">nevezetes azonosságot</a>, nevezetesen, hogy akármilyen A,B valós számokra <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)^{2}=A^{2}-2AB+B^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>B</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>A</mi> <mi>B</mi> <mo>+</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A-B)^{2}=A^{2}-2AB+B^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc72b1b51cdb8f4bc6ff3186ba6feeecc520ab71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.275ex; height:3.176ex;" alt="{\displaystyle (A-B)^{2}=A^{2}-2AB+B^{2}}"></span>, ennélfogva az A := c és B := x háromszög-oldalhosszakra <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 (c-x)^{2}=c^{2}-2cx+x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>c</mi> <mi>x</mi> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c-x)^{2}=c^{2}-2cx+x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cee3aaac74391c1800b6f1ad27132dc7a63e4a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.764ex; height:3.176ex;" alt="{\displaystyle (c-x)^{2}=c^{2}-2cx+x^{2}}"></span>; </p> <dl><dd><dl><dd><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 c^{2}-2cx+x^{2}+b^{2}-x^{2}=a^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>c</mi> <mi>x</mi> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}-2cx+x^{2}+b^{2}-x^{2}=a^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07fc0bfea33265226ab3f759d87d50b8139c8764" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:29.124ex; height:2.843ex;" alt="{\displaystyle c^{2}-2cx+x^{2}+b^{2}-x^{2}=a^{2}}"></span></dd></dl></dd></dl> <p>Az ellenkező előjellel szereplő x^{2}-es tagok kiejtik egymást, marad: </p> <dl><dd><dl><dd><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 c^{2}-2cx+x^{2}+b^{2}-x^{2}=a^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>c</mi> <mi>x</mi> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}-2cx+x^{2}+b^{2}-x^{2}=a^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07fc0bfea33265226ab3f759d87d50b8139c8764" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:29.124ex; height:2.843ex;" alt="{\displaystyle c^{2}-2cx+x^{2}+b^{2}-x^{2}=a^{2}}"></span></dd></dl></dd></dl> <p>Ebben az első ránézésre másodfokú egyenletben már csak egy ismeretlen szerepel, az x. Mivel ez egyetlen helyen fordul elő az egyenletben, és csak az első hatványon, a fönti egyenlet szerencsés módon valójában elsőfokú, ennélfogva az x ismeretlen mennyiség könnyedén kifejezhető, részint egy átrendezés, </p> <dl><dd><dl><dd><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^{2}+b^{2}+c^{2}=2cx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mi>c</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -a^{2}+b^{2}+c^{2}=2cx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83737d659c14fd3997d0de1494d0c237ef26ad34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.483ex; height:2.843ex;" alt="{\displaystyle -a^{2}+b^{2}+c^{2}=2cx}"></span></dd></dl></dd></dl> <p>részint 2c-vel való osztás után: </p> <dl><dd><dl><dd><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={\frac {-a^{2}+b^{2}+c^{2}}{2c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>c</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {-a^{2}+b^{2}+c^{2}}{2c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03755ad45ee3a8b0a87d72a459a81548805e44c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.15ex; height:5.676ex;" alt="{\displaystyle x={\frac {-a^{2}+b^{2}+c^{2}}{2c}}}"></span></dd></dl></dd></dl> <p>Visszahelyettesítve ezt az első egyenlet (*)-gal megjelölt formájába: </p> <dl><dd><dl><dd><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 m^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00d80831ded84ee5d9e1708e304c8868aa246409" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.095ex; height:2.676ex;" alt="{\displaystyle m^{2}}"></span>   <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 =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span> <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 b^{2}-x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{2}-x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da550bc374906f73a9953e7a46a28bb484b24e16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.276ex; height:2.843ex;" alt="{\displaystyle b^{2}-x^{2}}"></span> <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 =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span>   <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 b^{2}-\left({\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>c</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{2}-\left({\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c31f4e05676324c84a72c5dcd4e387d58c67005" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.09ex; height:6.676ex;" alt="{\displaystyle b^{2}-\left({\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)^{2}}"></span>   <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 =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span>   <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 b^{2}-{\frac {\left(-a^{2}+b^{2}+c^{2}\right)^{2}}{\left(2c\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{2}-{\frac {\left(-a^{2}+b^{2}+c^{2}\right)^{2}}{\left(2c\right)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77ac772b7b7712ca6ac3deebec97084a8d83871c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.798ex; height:7.843ex;" alt="{\displaystyle b^{2}-{\frac {\left(-a^{2}+b^{2}+c^{2}\right)^{2}}{\left(2c\right)^{2}}}}"></span>   <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 =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span>   <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 b^{2}-{\frac {\left(-a^{2}+b^{2}+c^{2}\right)^{2}}{4c^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>4</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{2}-{\frac {\left(-a^{2}+b^{2}+c^{2}\right)^{2}}{4c^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c8e04e0d726c187ca970a7001421a37c1240d19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:22.798ex; height:6.843ex;" alt="{\displaystyle b^{2}-{\frac {\left(-a^{2}+b^{2}+c^{2}\right)^{2}}{4c^{2}}}}"></span>   <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 =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span> <br /> <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 =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span>   <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 {\frac {4c^{2}\cdot b^{2}}{4c^{2}}}-{\frac {\left(-a^{2}+b^{2}+c^{2}\right)^{2}}{4c^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>4</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4c^{2}\cdot b^{2}}{4c^{2}}}-{\frac {\left(-a^{2}+b^{2}+c^{2}\right)^{2}}{4c^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57f2bedcdd60ab1a7aca0089c7aa3ef8e4eb8d83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:28.537ex; height:6.843ex;" alt="{\displaystyle {\frac {4c^{2}\cdot b^{2}}{4c^{2}}}-{\frac {\left(-a^{2}+b^{2}+c^{2}\right)^{2}}{4c^{2}}}}"></span>   <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 =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span>   <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 {\frac {4b^{2}c^{2}-\left(-a^{2}+b^{2}+c^{2}\right)^{2}}{4c^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4b^{2}c^{2}-\left(-a^{2}+b^{2}+c^{2}\right)^{2}}{4c^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44bd1543913b7a26cee1cf6430c8d85889ad5a08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:26.022ex; height:6.843ex;" alt="{\displaystyle {\frac {4b^{2}c^{2}-\left(-a^{2}+b^{2}+c^{2}\right)^{2}}{4c^{2}}}}"></span>   <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 =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span>   <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 {\frac {\left(2bc\right)^{2}-\left(-a^{2}+b^{2}+c^{2}\right)^{2}}{\left(2c\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>b</mi> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\left(2bc\right)^{2}-\left(-a^{2}+b^{2}+c^{2}\right)^{2}}{\left(2c\right)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/086e9a83661689eceff5f081b12fc1be16661956" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.777ex; height:7.843ex;" alt="{\displaystyle {\frac {\left(2bc\right)^{2}-\left(-a^{2}+b^{2}+c^{2}\right)^{2}}{\left(2c\right)^{2}}}}"></span>   <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 =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span>   <br /> <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 =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span> <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 {\frac {\left(2bc\right)^{2}-\left(-a^{2}+b^{2}+c^{2}\right)^{2}}{\left(2c\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>b</mi> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\left(2bc\right)^{2}-\left(-a^{2}+b^{2}+c^{2}\right)^{2}}{\left(2c\right)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/086e9a83661689eceff5f081b12fc1be16661956" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.777ex; height:7.843ex;" alt="{\displaystyle {\frac {\left(2bc\right)^{2}-\left(-a^{2}+b^{2}+c^{2}\right)^{2}}{\left(2c\right)^{2}}}}"></span>.</dd></dl></dd></dl> <p>Alkalmazva az <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^{2}-B^{2}=\left(A+B\right)\left(A-B\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>A</mi> <mo>+</mo> <mi>B</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>A</mi> <mo>−</mo> <mi>B</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{2}-B^{2}=\left(A+B\right)\left(A-B\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a95dd576ab4eee3225fb77c214465cb91e7e3315" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.255ex; height:3.176ex;" alt="{\displaystyle A^{2}-B^{2}=\left(A+B\right)\left(A-B\right)}"></span>   „nevezetes azonosságot” az <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=2bc}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>2</mn> <mi>b</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=2bc}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66c88de309c01f9f159f2663e38b1aadc74c1106" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.008ex; height:2.176ex;" alt="{\displaystyle A=2bc}"></span> és <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 B=-a^{2}+b^{2}+c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=-a^{2}+b^{2}+c^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2b1f58fdbb3e8a861da285250e78db3192f577a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.748ex; height:2.843ex;" alt="{\displaystyle B=-a^{2}+b^{2}+c^{2}}"></span> esetekre; </p> <dl><dd><dl><dd><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 m^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00d80831ded84ee5d9e1708e304c8868aa246409" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.095ex; height:2.676ex;" alt="{\displaystyle m^{2}}"></span>   <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 =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span>   <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 {\frac {\left[\left(2bc\right)+\left(-a^{2}+b^{2}+c^{2}\right)\right]\left[\left(2bc\right)-\left(-a^{2}+b^{2}+c^{2}\right)\right]}{\left(2c\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow> <mo>[</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>b</mi> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow> <mo>[</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>b</mi> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\left[\left(2bc\right)+\left(-a^{2}+b^{2}+c^{2}\right)\right]\left[\left(2bc\right)-\left(-a^{2}+b^{2}+c^{2}\right)\right]}{\left(2c\right)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13dd1735dba6d6084f53f5d7da3686ef48588dd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:52.766ex; height:7.343ex;" alt="{\displaystyle {\frac {\left[\left(2bc\right)+\left(-a^{2}+b^{2}+c^{2}\right)\right]\left[\left(2bc\right)-\left(-a^{2}+b^{2}+c^{2}\right)\right]}{\left(2c\right)^{2}}}}"></span>   <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 =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span>   <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 {\frac {\left(-a^{2}+b^{2}+2bc+c^{2}\right)\left(2bc+a^{2}-b^{2}-c^{2}\right)}{\left(2c\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>b</mi> <mi>c</mi> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>b</mi> <mi>c</mi> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\left(-a^{2}+b^{2}+2bc+c^{2}\right)\left(2bc+a^{2}-b^{2}-c^{2}\right)}{\left(2c\right)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28eef30b0933374be05e50a0387b2740fb218ef7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:43.461ex; height:7.343ex;" alt="{\displaystyle {\frac {\left(-a^{2}+b^{2}+2bc+c^{2}\right)\left(2bc+a^{2}-b^{2}-c^{2}\right)}{\left(2c\right)^{2}}}}"></span>   <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 =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span> <br /> <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 =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span>   <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 {\frac {\left(-a^{2}+b^{2}+2bc+c^{2}\right)\left(a^{2}-b^{2}+2bc-c^{2}\right)}{\left(2c\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>b</mi> <mi>c</mi> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>b</mi> <mi>c</mi> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\left(-a^{2}+b^{2}+2bc+c^{2}\right)\left(a^{2}-b^{2}+2bc-c^{2}\right)}{\left(2c\right)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc4c8b32622446c601d9f60f8188247c4a632ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:43.461ex; height:7.343ex;" alt="{\displaystyle {\frac {\left(-a^{2}+b^{2}+2bc+c^{2}\right)\left(a^{2}-b^{2}+2bc-c^{2}\right)}{\left(2c\right)^{2}}}}"></span>   <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 =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span> <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 {\frac {\left[-a^{2}+\left(b+c\right)^{2}\right]\left[a^{2}-\left(b^{2}-2bc+c^{2}\right)\right]}{\left(2c\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>b</mi> <mo>+</mo> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mrow> <mo>[</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>b</mi> <mi>c</mi> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\left[-a^{2}+\left(b+c\right)^{2}\right]\left[a^{2}-\left(b^{2}-2bc+c^{2}\right)\right]}{\left(2c\right)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5526118580660334f8c8c0c4b272cd3dbad6bce1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:40.213ex; height:8.843ex;" alt="{\displaystyle {\frac {\left[-a^{2}+\left(b+c\right)^{2}\right]\left[a^{2}-\left(b^{2}-2bc+c^{2}\right)\right]}{\left(2c\right)^{2}}}}"></span>   <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 =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span> <br /> <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 =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span>   <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 {\frac {\left[-a^{2}+\left(b+c\right)^{2}\right]\left[a^{2}-\left(b-c\right)^{2}\right]}{\left(2c\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>b</mi> <mo>+</mo> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mrow> <mo>[</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>b</mi> <mo>−</mo> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\left[-a^{2}+\left(b+c\right)^{2}\right]\left[a^{2}-\left(b-c\right)^{2}\right]}{\left(2c\right)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3822ab0d247bcc8e040f86e053af09ca29390a82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.087ex; height:8.843ex;" alt="{\displaystyle {\frac {\left[-a^{2}+\left(b+c\right)^{2}\right]\left[a^{2}-\left(b-c\right)^{2}\right]}{\left(2c\right)^{2}}}}"></span>   <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 =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span>   <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 {\frac {\left[\left(b+c\right)^{2}-a^{2}\right]\left[a^{2}-\left(b-c\right)^{2}\right]}{\left(2c\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow> <mo>[</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>b</mi> <mo>+</mo> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mrow> <mo>[</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>b</mi> <mo>−</mo> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\left[\left(b+c\right)^{2}-a^{2}\right]\left[a^{2}-\left(b-c\right)^{2}\right]}{\left(2c\right)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fb7a53722362bf9b281bd76927e94db1ab8ef25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.278ex; height:8.843ex;" alt="{\displaystyle {\frac {\left[\left(b+c\right)^{2}-a^{2}\right]\left[a^{2}-\left(b-c\right)^{2}\right]}{\left(2c\right)^{2}}}}"></span> .</dd></dl></dd></dl> <p>Az utolsó két átalakításnál a két tag összegének négyzetére vonatkozó <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 B^{2}+2BC+C^{2}\ =\ \left(B+C\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>B</mi> <mi>C</mi> <mo>+</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <msup> <mrow> <mo>(</mo> <mrow> <mi>B</mi> <mo>+</mo> <mi>C</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B^{2}+2BC+C^{2}\ =\ \left(B+C\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bca40db5b03a49c7ba08495a92502bb91d27b211" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.538ex; height:3.343ex;" alt="{\displaystyle B^{2}+2BC+C^{2}\ =\ \left(B+C\right)^{2}}"></span>, szintén nevezetes azonosságot alkalmaztuk (az egyik zárójelen belül), illetve egy harmadikat, a két tag különbségére vonatkozót, amely a következőképp fest: <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 B^{2}-2BC+C^{2}\ =\ \left(B-C\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>B</mi> <mi>C</mi> <mo>+</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <msup> <mrow> <mo>(</mo> <mrow> <mi>B</mi> <mo>−</mo> <mi>C</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B^{2}-2BC+C^{2}\ =\ \left(B-C\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fc72a9b77b8b5905045471be65b4a2dd5a2b11a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.538ex; height:3.343ex;" alt="{\displaystyle B^{2}-2BC+C^{2}\ =\ \left(B-C\right)^{2}}"></span>. </p><p>Megint csak a már említett <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^{2}-B^{2}\ =\ \left(A+B\right)\left(A-B\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow> <mo>(</mo> <mrow> <mi>A</mi> <mo>+</mo> <mi>B</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>A</mi> <mo>−</mo> <mi>B</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{2}-B^{2}\ =\ \left(A+B\right)\left(A-B\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8059c85ede9e4b32e3009fb3dc381af5cc542a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.804ex; height:3.176ex;" alt="{\displaystyle A^{2}-B^{2}\ =\ \left(A+B\right)\left(A-B\right)}"></span> azonosságot alkalmazva a fentebbi kifejezés szögletes zárójelbe rakott részkifejezéseire (az első esetében az <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\ :=\ \left(b+c\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mtext> </mtext> <mo>:=</mo> <mtext> </mtext> <mrow> <mo>(</mo> <mrow> <mi>b</mi> <mo>+</mo> <mi>c</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\ :=\ \left(b+c\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fef2eacb3a8f7d1a6442ed5cabb7df518367970a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.691ex; height:2.843ex;" alt="{\displaystyle A\ :=\ \left(b+c\right)}"></span>   és <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 B\ :=\ a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mtext> </mtext> <mo>:=</mo> <mtext> </mtext> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\ :=\ a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/208157c09c6150f3e2d11a3a3951c2b8be76e23f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.9ex; height:2.176ex;" alt="{\displaystyle B\ :=\ a}"></span>; míg a második esetében az <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\ :=\ a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mtext> </mtext> <mo>:=</mo> <mtext> </mtext> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\ :=\ a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/861bfb050090f203b97805dae143954fe5a2cbdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.88ex; height:2.176ex;" alt="{\displaystyle A\ :=\ a}"></span> és <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 B\ :=\ \left(b-c\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mtext> </mtext> <mo>:=</mo> <mtext> </mtext> <mrow> <mo>(</mo> <mrow> <mi>b</mi> <mo>−</mo> <mi>c</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\ :=\ \left(b-c\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6f26008d67fb620abfce10f44406e5aa2bbc30a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.712ex; height:2.843ex;" alt="{\displaystyle B\ :=\ \left(b-c\right)}"></span> helyettesítésekkel): </p> <dl><dd><dl><dd><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 m^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00d80831ded84ee5d9e1708e304c8868aa246409" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.095ex; height:2.676ex;" alt="{\displaystyle m^{2}}"></span> <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 =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span> <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 {\frac {\left[\left(b+c+a\right)\left(b+c-a\right)\right]\cdot \left[\left(a+(b-c)\right)\left(a-(b-c)\right)\right]}{\left(2c\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow> <mo>[</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo>+</mo> <mi>a</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo>−</mo> <mi>a</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>⋅</mo> <mrow> <mo>[</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\left[\left(b+c+a\right)\left(b+c-a\right)\right]\cdot \left[\left(a+(b-c)\right)\left(a-(b-c)\right)\right]}{\left(2c\right)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d37edc6a1fae8d46f2df5ca060fa0ced4721931" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; 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