Solution 2.3:1c

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m (Lösning 2.3:1c moved to Solution 2.3:1c: Robot: moved page)
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As always when completing the square, we focus on the quadratic and linear terms
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<center> [[Image:2_3_1c.gif]] </center>
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<math>2x-x^{2}</math>
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, which we also can write as
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<math>-\left( x^{2}-2x \right)</math>
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. If we neglect the minus sign, we can complete square of the expression
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<math>2x-x^{2}</math>
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by using the formula
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<math>x^{2}-ax=\left( x-\frac{a}{2} \right)^{2}-\left( \frac{a}{2} \right)^{2}</math>
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and we obtain
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<math>x^{2}-2x=\left( x-\frac{2}{2} \right)^{2}-\left( \frac{2}{2} \right)^{2}=\left( x-1 \right)^{2}-1</math>
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This means that
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<math>\begin{align}
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& 5+2x-x^{2}=5-\left( x^{2}-2x \right)=5-\left( \left( x-1 \right)^{2}-1 \right) \\
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& \\
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& =5-\left( x-1 \right)^{2}+1=6-\left( x-1 \right)^{2} \\
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& \\
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\end{align}</math>
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A quick check shows that we have completed the square correctly.:
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<math>\begin{align}
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& 6-\left( x-1 \right)^{2}=6-\left( x^{2}-2x+1 \right)=6-x^{2}+2x-1 \\
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& \\
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& =5+2x-x^{2} \\
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& \\
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\end{align}</math>

Revision as of 11:19, 12 September 2008

As always when completing the square, we focus on the quadratic and linear terms \displaystyle 2x-x^{2} , which we also can write as \displaystyle -\left( x^{2}-2x \right) . If we neglect the minus sign, we can complete square of the expression \displaystyle 2x-x^{2} by using the formula


\displaystyle x^{2}-ax=\left( x-\frac{a}{2} \right)^{2}-\left( \frac{a}{2} \right)^{2}


and we obtain


\displaystyle x^{2}-2x=\left( x-\frac{2}{2} \right)^{2}-\left( \frac{2}{2} \right)^{2}=\left( x-1 \right)^{2}-1


This means that


\displaystyle \begin{align} & 5+2x-x^{2}=5-\left( x^{2}-2x \right)=5-\left( \left( x-1 \right)^{2}-1 \right) \\ & \\ & =5-\left( x-1 \right)^{2}+1=6-\left( x-1 \right)^{2} \\ & \\ \end{align}


A quick check shows that we have completed the square correctly.:


\displaystyle \begin{align} & 6-\left( x-1 \right)^{2}=6-\left( x^{2}-2x+1 \right)=6-x^{2}+2x-1 \\ & \\ & =5+2x-x^{2} \\ & \\ \end{align}

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