For how many real numbers x is √-(x+1)^2 a real number?

(D) a finite number greater than two

1976 AMC 12 Problem 2

Attempt Problem 2 of the 1976 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1976 AMC 12 solutions, or check the answer key.

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For how many real numbers

x

\sqrt{-(x+1)^2}

a real number?

none

one

two

a finite number greater than two

infinitely many

A real square root requires a nonnegative radicand

The quantity

-(x+1)^2

can be nonnegative only when the square vanishes

Since

(x+1)^2\ge0,

its negative is at most

0.

It is nonnegative only when

(x+1)^2=0,

which occurs for the single value

x=-1.

Therefore, the correct answer is B.