Difference between revisions of "1977 AHSME Problems/Problem 21"
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For how many values of the coefficient a do the equations <cmath>\begin{align*}x^2+ax+1=0 \\ x^2-x-a=0\end{align*}</cmath> have a common real solution? | For how many values of the coefficient a do the equations <cmath>\begin{align*}x^2+ax+1=0 \\ x^2-x-a=0\end{align*}</cmath> have a common real solution? | ||
− | + | <math>\textbf{(A)}\ 0 \qquad | |
\textbf{(B)}\ 1 \qquad | \textbf{(B)}\ 1 \qquad | ||
\textbf{(C)}\ 2 \qquad | \textbf{(C)}\ 2 \qquad | ||
\textbf{(D)}\ 3 \qquad | \textbf{(D)}\ 3 \qquad | ||
− | \textbf{(E)}\ \infty | + | \textbf{(E)}\ \infty</math> |
+ | |||
+ | ==Solution== | ||
+ | |||
+ | Subtracting the equations, we get <math>ax+x+1+a=0</math>, or <math>(x+1)(a+1)=0</math>, so <math>x=-1</math> or <math>a=-1</math>. If <math>x=-1</math>, then <math>a=2</math>, which satisfies the condition. If <math>a=-1</math>, then <math>x</math> is nonreal. This means that <math>a=-1</math> is the only number that works, so our answer is <math>(B)</math>. | ||
+ | |||
+ | ~alexanderruan |
Latest revision as of 00:22, 2 January 2024
Problem 21
For how many values of the coefficient a do the equations have a common real solution?
Solution
Subtracting the equations, we get , or
, so
or
. If
, then
, which satisfies the condition. If
, then
is nonreal. This means that
is the only number that works, so our answer is
.
~alexanderruan