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Let r and s be the roots of y^2-19y+7. Find (r-2)(s-2)

Apr 9, 2018

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Let r and s be the roots of y^2-19y+7. Find (r-2)(s-2)

$$\small{ \begin{array}{|lrcll|} \hline & (y-r)(y-s) &=& y^2-19y+7 \\ & y^2-y(r+s) +rs &=& y^2-19y+7 \quad & | \quad (r-2)(s-2) = 4-2(r+s)+rs \\ y=2:& 2^2-2(r+s) +rs &=& 2^2-19\cdot 2+7 \\ & 4-2(r+s) +rs &=& 4-38 +7 \quad & | \quad 4-2(r+s)+rs = (r-2)(s-2) \\ & (r-2)(s-2) &=& 4-38 +7 \\ & (r-2)(s-2) &=& 11-38 \\ & \mathbf{ (r-2)(s-2) } & \mathbf{=} & \mathbf{ -27 } \\ \hline \end{array} }$$

Apr 9, 2018
edited by heureka  Apr 9, 2018