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<title>A.II.3.26</title>
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<div class="feladat">
<b>Feladat: 3.26.</b><br /> <a name="Komal_C_366_1995_2_84" /><a href="bib_box.php?mode=sne-s-j-&amp;citation_num=" target="bib_box" onclick="mutat('bib_box.php?mode=sne-s-j-&amp;citation_num='); return false;"></a>Oldjuk meg a következő egyenletet a valós számok halmazán:

<table width="100%"><tr><td align="center">

<br />

<table width="100%"><tr><td align="center">

    <m:math xmlns="http://www.w3.org/1998/Math/MathML">

    <m:mstyle displaystyle="true"><m:mrow><m:mn>1</m:mn><m:mo>+</m:mo><m:mn>2</m:mn><m:msqrt><m:mrow><m:mi>x</m:mi></m:mrow></m:msqrt><m:mo>-</m:mo><m:mroot><m:mrow><m:mi>x</m:mi></m:mrow><m:mrow><m:mn>3</m:mn></m:mrow></m:mroot><m:mo>-</m:mo><m:mn>2</m:mn><m:mroot><m:mrow><m:mi>x</m:mi></m:mrow><m:mrow><m:mn>6</m:mn></m:mrow></m:mroot><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow>

    </m:mstyle></m:math>

</td></tr></table>

<br />

<a name="eq:Komal_C_366_1995_2_84" /></td><td width="1">(1)</td></tr></table>

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<div class="feladat">
<a name="_solution_Komal_C_366_1995_2_84" /><b>Megoldás: 3.26</b><br />
Alkalmazva a 

<m:math xmlns="http://www.w3.org/1998/Math/MathML">

<m:mrow><m:mroot><m:mrow><m:mi>x</m:mi></m:mrow><m:mrow><m:mn>6</m:mn></m:mrow></m:mroot><m:mo>=</m:mo><m:mi>y</m:mi></m:mrow></m:math> helyettesítést, a kapott harmadfokú egyenlet <m:math xmlns="http://www.w3.org/1998/Math/MathML">

<m:mrow><m:mn>2</m:mn>

<m:msup><m:mrow><m:mi>y</m:mi></m:mrow><m:mrow><m:mn>3</m:mn></m:mrow>

</m:msup>

<m:mo>-</m:mo>

<m:msup><m:mrow><m:mi>y</m:mi></m:mrow><m:mrow><m:mn>2</m:mn></m:mrow>

</m:msup>

<m:mo>-</m:mo><m:mn>2</m:mn><m:mi>y</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow></m:math>, ennek keressük először egész gyökét. Szóba jön <m:math xmlns="http://www.w3.org/1998/Math/MathML">

<m:mrow><m:mo>&PlusMinus;</m:mo><m:mn>1</m:mn></m:mrow></m:math>, ellenőrizve mindkettő jó, a három gyök  <m:math xmlns="http://www.w3.org/1998/Math/MathML">

<m:mrow><m:mo>-</m:mo><m:mn>1</m:mn><m:mo>;</m:mo><m:mi>&ensp;</m:mi><m:mn>1</m:mn><m:mo stretchy="false">/</m:mo><m:mn>2</m:mn><m:mo>;</m:mo><m:mi>&ensp;</m:mi><m:mn>1</m:mn></m:mrow></m:math>. Ezeket visszaírva <m:math xmlns="http://www.w3.org/1998/Math/MathML">

<m:mrow>

<m:msub><m:mrow><m:mi>x</m:mi></m:mrow><m:mrow><m:mn>1</m:mn></m:mrow>

</m:msub>

<m:mo>=</m:mo><m:mn>1</m:mn><m:mo>,</m:mo><m:mi>&ensp;</m:mi>

<m:msub><m:mrow><m:mi>x</m:mi></m:mrow><m:mrow><m:mn>2</m:mn></m:mrow>

</m:msub>

<m:mo>=</m:mo><m:mn>1</m:mn><m:mo stretchy="false">/</m:mo><m:mn>64</m:mn></m:mrow></m:math>. (

<m:math xmlns="http://www.w3.org/1998/Math/MathML">

<m:mrow><m:mroot><m:mrow><m:mi>x</m:mi></m:mrow><m:mrow><m:mn>6</m:mn></m:mrow></m:mroot><m:mo>=</m:mo><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:math> lehetetlen.)
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