Lösung 2.3:4c

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The equation
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<center> [[Image:2_3_4c.gif]] </center>
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<math>\left( x-\text{3} \right)\left( x-\sqrt{\text{3}} \right)=0</math>
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is a second-degree equation which has
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<math>x=\text{3 }</math>
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and
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<math>x=\sqrt{\text{3}}</math>
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as roots; when
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<math>x=\text{3 }</math>, the first factor is zero and when
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<math>x=\sqrt{\text{3}}</math>
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the second factor is zero.
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If we expand the equations left-hand side, we get the equation in standard form,
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<math>\begin{align}
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& \left( x-\text{3} \right)\left( x-\sqrt{\text{3}} \right)=x^{2}-\sqrt{\text{3}}x-3x+3\sqrt{\text{3}} \\
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& =x^{2}-\left( 3+\sqrt{\text{3}} \right)x+3\sqrt{\text{3}}=0 \\
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\end{align}</math>
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NOTE: the general answer is,
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<math>ax^{2}-\left( 3+\sqrt{\text{3}} \right)ax+3\sqrt{\text{3}}a=0</math>
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where
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<math>a\ne 0</math>
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is a constant.

Version vom 09:38, 21. Sep. 2008

The equation \displaystyle \left( x-\text{3} \right)\left( x-\sqrt{\text{3}} \right)=0 is a second-degree equation which has \displaystyle x=\text{3 } and \displaystyle x=\sqrt{\text{3}} as roots; when \displaystyle x=\text{3 }, the first factor is zero and when \displaystyle x=\sqrt{\text{3}} the second factor is zero.

If we expand the equations left-hand side, we get the equation in standard form,


\displaystyle \begin{align} & \left( x-\text{3} \right)\left( x-\sqrt{\text{3}} \right)=x^{2}-\sqrt{\text{3}}x-3x+3\sqrt{\text{3}} \\ & =x^{2}-\left( 3+\sqrt{\text{3}} \right)x+3\sqrt{\text{3}}=0 \\ \end{align}


NOTE: the general answer is,


\displaystyle ax^{2}-\left( 3+\sqrt{\text{3}} \right)ax+3\sqrt{\text{3}}a=0


where \displaystyle a\ne 0 is a constant.

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse1-TU-Berlin/index.php/L%C3%B6sung_2.3:4c“