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Difference between revisions of "1977 AHSME Problems/Problem 21"

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== Problem 21 ==
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== 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?
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\textbf{(E)}\ \infty</math>
 
\textbf{(E)}\ \infty</math>
  
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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 a=-1 is the only number that works, so our answer is <math>(B) 1</math>.
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==Solution==
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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>.
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~alexanderruan
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== See Also ==
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{{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 \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 $(B)$.

~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
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