https://wikiskola.se/index.php?action=history&feed=atom&title=Ekvationer_med_en_k%C3%A4nd_reel_rot 2026-05-21T13:24:10Z Versionshistorik för denna sida på wikin MediaWiki 1.41.1 https://wikiskola.se/index.php?title=Ekvationer_med_en_k%C3%A4nd_reel_rot&diff=38167&oldid=prev 2016-12-06T13:16:40Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="sv"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Äldre version</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Versionen från 6 december 2016 kl. 13.16</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Rad 1:</td> <td colspan="2" class="diff-lineno">Rad 1:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Matematik]] [[Category:Ma4]]  [[Category:Aritmetik, algebra och geometri]]  [[Category:Komplexa tal]]  </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Matematik]] [[Category:Ma4]]  [[Category:Aritmetik, algebra och geometri]]  [[Category:Komplexa tal]]  </div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{Lm4 |Ekvationer med en känd reell rot|211-213}}</ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{Lm4 |Ekvationer med </del>en <del style="font-weight: bold; text-decoration: none;">känd reell </del>rot<del style="font-weight: bold; text-decoration: none;">|211</del>-<del style="font-weight: bold; text-decoration: none;">213}}</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Nu när vi kan faktorssatsen och polynomdivision kan vi lösa tredjegradsekvationer. gången är som följer:</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"># Hitta </ins>en rot <ins style="font-weight: bold; text-decoration: none;">x=a till funktionen p(x). Ibland är den given i uppgiften men du kan gissa den eller rita grafen och se var kurvan skär x-axeln.</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"># (x-a) är en faktor så att p(x) = (x-a) q(x)</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"># Polynomdivision ger q(x) = p(x) / (x-a)</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"># Sätt q(x) = 0 och lös med pq</ins>-<ins style="font-weight: bold; text-decoration: none;">formeln ger de två andra (komplexa) röterna.</ins></div></td></tr> </table>

Hakan https://wikiskola.se/index.php?title=Ekvationer_med_en_k%C3%A4nd_reel_rot&diff=35015&oldid=prev 2016-03-01T15:37:30Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="sv"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Äldre version</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Versionen från 1 mars 2016 kl. 15.37</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Rad 1:</td> <td colspan="2" class="diff-lineno">Rad 1:</td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Matematik]] [[Category:Ma4]]  [[Category:Aritmetik, algebra och geometri]]  [[Category:Komplexa tal]] </ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{Lm4 |Ekvationer med en känd reell rot|211-213}}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{Lm4 |Ekvationer med en känd reell rot|211-213}}</div></td></tr> </table>

Hakan https://wikiskola.se/index.php?title=Ekvationer_med_en_k%C3%A4nd_reel_rot&diff=31824&oldid=prev 2015-04-22T22:08:51Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="sv"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Äldre version</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Versionen från 22 april 2015 kl. 22.08</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Rad 1:</td> <td colspan="2" class="diff-lineno">Rad 1:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{Lm4 |Ekvationer med en känd reell rot|211-213}}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{Lm4 |Ekvationer med en känd reell rot|211-213}}</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2" class="diff-side-added"></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{#ev:youtube | v=4IEVzPsN5_E | 340 | right|Johan Sandell, TIS Lund}}</del></div></td><td colspan="2" class="diff-side-added"></td></tr> </table>

Hakan https://wikiskola.se/index.php?title=Ekvationer_med_en_k%C3%A4nd_reel_rot&diff=31812&oldid=prev 2015-04-21T22:50:58Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="sv"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Äldre version</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Versionen från 21 april 2015 kl. 22.50</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Rad 1:</td> <td colspan="2" class="diff-lineno">Rad 1:</td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{{<del style="font-weight: bold; text-decoration: none;">LM4 </del>|Ekvationer med en känd reell rot|211-<del style="font-weight: bold; text-decoration: none;">212</del>}}</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{{<ins style="font-weight: bold; text-decoration: none;">Lm4 </ins>|Ekvationer med en känd reell rot|211-<ins style="font-weight: bold; text-decoration: none;">213</ins>}}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{#ev:youtube | v=4IEVzPsN5_E | 340 | right|Johan Sandell, TIS Lund}}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{#ev:youtube | v=4IEVzPsN5_E | 340 | right|Johan Sandell, TIS Lund}}</div></td></tr> </table>

Hakan https://wikiskola.se/index.php?title=Ekvationer_med_en_k%C3%A4nd_reel_rot&diff=31811&oldid=prev 2015-04-21T22:50:31Z

<p>Skapade sidan med &#039;{{LM4 |Ekvationer med en känd reell rot|211-212}} {{#ev:youtube | v=4IEVzPsN5_E | 340 | right|Johan Sandell, TIS Lund}}&#039;</p> <p><b>Ny sida</b></p><div>{{LM4 |Ekvationer med en känd reell rot|211-212}}<br /> <br /> {{#ev:youtube | v=4IEVzPsN5_E | 340 | right|Johan Sandell, TIS Lund}}</div>

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