1958 AHSME Problems/Problem 10

Problem

For what real values of $k$ , other than $k \equal{} 0$ (Error compiling LaTeX. Unknown error_msg), does the equation $x^2 \plus{} kx \plus{} k^2 \equal{} 0$ (Error compiling LaTeX. Unknown error_msg) have real roots?

$\textbf{(A)}\ {k < 0}\qquad \textbf{(B)}\ {k > 0} \qquad \textbf{(C)}\ {k \ge 1} \qquad \textbf{(D)}\ \text{all values of }{k}\qquad \textbf{(E)}\ \text{no values of }{k}$

Solution

An expression of the form $ax^2+bx+c$ has at least one real root when $b^2-4ac \geq 0$ .

Substituting $k$ for $b$ and $k^2$ for $c$ , we have

$k^2-4k^2 \geq 0$

$-3k^2 \geq 0$

but the range of $-3k^2$ is $(-\infty,0]$ , so the answer is $\boxed{\text{(E)}}$

1958 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
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1958 AHSME Problems/Problem 10

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