1960 AHSME Problems/Problem 11

Problem

For a given value of $k$ the product of the roots of $x^2-3kx+2k^2-1=0$ is $7$ . The roots may be characterized as:

$\textbf{(A) }\text{integral and positive} \qquad\textbf{(B) }\text{integral and negative} \qquad \\ \textbf{(C) }\text{rational, but not integral} \qquad\textbf{(D) }\text{irrational} \qquad\textbf{(E) } \text{imaginary}$

Solution

If the product of the roots are $7$ , then by Vieta's formulas, $2k^2-1=7$ Solve for $k$ in the resulting equation to get $2k^2=8$ $k^2=4$ $k=\pm 2$ That means the two quadratics are $x^2-6x+7=0$ and $x^2+6x+7=0$ . Since $b^2$ , $a$ , and $c$ are the same, the discriminant of both is $36-(4 \cdot 1 \cdot 7) = 8$ . Because $8$ is not a perfect square, the roots for both are irrational, so the answer is $\boxed{\textbf{(D)}}$ .

1960 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
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1960 AHSME Problems/Problem 11

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