1952 AHSME Problems/Problem 6

Problem

The difference of the roots of $x^2-7x-9=0$ is:

$\textbf{(A) \ }+7 \qquad \textbf{(B) \ }+\frac{7}{2} \qquad \textbf{(C) \ }+9 \qquad \textbf{(D) \ }2\sqrt{85} \qquad \textbf{(E) \ }\sqrt{85}$

Solution

Denote the $2$ roots of this quadratic as $r_1$ and $r_2$ . Note that $(r_1-r_2)^2=(r_1+r_2)^2-4r_1r_2$ . By Vieta's Formula's, $r_1+r_2=7$ , and $r_1r_2=-9$ . Thus, $r_1-r_2=\sqrt{49+4\cdot 9}=\boxed{\textbf{(E)}\ \sqrt{85}}$ .

1952 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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1952 AHSME Problems/Problem 6

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