# Difference between revisions of "1958 AHSME Problems/Problem 10"

## Problem

For what real values of $k$, other than $k = 0$, does the equation $x^2 + kx + k^2 = 0$ 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 $\boxed{\text{(E)}}$