1977 AHSME Problems/Problem 21

Problem 21

For how many values of the coefficient a do the equations $\begin{align*}x^2+ax+1=0 \\ x^2-x-a=0\end{align*}$ have a common real solution?

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \infty$

Solution

Subtracting the equations, we get $ax+x+1+a=0$ , or $(x+1)(a+1)=0$ , so $x=-1$ or $a=-1$ . If $x=-1$ , then $a=2$ , which satisfies the condition. If $a=-1$ , then $x$ is nonreal. This means that $a=-1$ is the only number that works, so our answer is $\textbf{(B)}\ 1 \qquad$ .

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1977 AHSME Problems/Problem 21

Problem 21

Solution