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Let $a$ and $b$ be the roots of $y^2 + 5y - 11 = 0$. Find $(a + 3)(b + 3).$

 Feb 25, 2020
 #1
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The roots are (-5 - sqrt(73))/2 and (-5 + sqrt(73))/2, so (a + 3)(b + 3) = ((-5 - sqrt(73))/2 + 3)((-5 + sqrt(73))/2 + 3) = -37/2.

 Feb 25, 2020
 #2
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Let \(a\) and \(b\) be the roots of \(y^2 + 5y - 11 = 0\).
Find \((a + 3)(b + 3)\).

 

Vieta's formulas:  \(\begin{array}{|rcll|} \hline y^2 \underbrace{+5}_{=-(a+b)}y \underbrace{- 11}_{=ab} &=& 0 \\ \hline \end{array} \)

 

\(\begin{array}{|rcll|} \hline ab&=& -11 \\ a+b &=& -5 \\\\ && \mathbf{(a+3)(b+3)} \\ &=& ab+3a+3b+9 \\ &=& ab+3(a+b) +9 \\ &=& -11+3(-5) + 9 \\ &=& -11-15+9 \\ &=& \mathbf{-17} \\ \hline \end{array}\)

 

laugh

 Feb 25, 2020

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