Difference between revisions of "1977 AHSME Problems/Problem 21"
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| − | == Problem | + | == Problem== |
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 | ||
| + | |||
| + | == See Also == | ||
| + | {{AHSME box|year=1977|num-b=20|num-a=22}} | ||
Latest revision as of 16:57, 17 December 2024
Problem
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
See Also
| 1977 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 20 |
Followed by Problem 22 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
