For how many values of the coefficient a do the equations x^2+ax+1&=0, x^2-x-a&=0 have a common real solution?

1977 AMC 12 Problem 21

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21.

For how many values of the coefficient

a

do the equations

\begin{aligned} x^2+ax+1&=0,\\ x^2-x-a&=0 \end{aligned}

have a common real solution?

0

1

2

3

infinitely many

Subtract the two equations to obtain a factored condition

Check separately the cases

a=-1

and

x=-1

Subtracting the second equation from the first gives

\begin{gathered} (a+1)x+(a+1)\\ =(a+1)(x+1)=0. \end{gathered}

a=-1,

the common equation is

x^2-x+1=0,

which has no real roots. Otherwise

x=-1,

and substitution into

x^2-x-a=0

gives

a=2.

This value works, so exactly one value of

a

qualifies.

Therefore, the correct answer is B.