1976 AMC 12 第 2 题

Question

有多少个实数

x

能使

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

为实数？

For how many real numbers

x

is

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

a real number?

没有

none

一个

one

两个

two

大于两个的有限个

a finite number greater than two

无穷多个

infinitely many

平方根在实数范围内有意义时，被开方数必须非负

A real square root requires a nonnegative radicand

-(x+1)^2

只有在这个平方为零时才可能非负

The quantity

-(x+1)^2

can be nonnegative only when the square vanishes

因为

(x+1)^2\ge0

，所以它的相反数至多为

0

。它只有在

(x+1)^2=0

时非负，而这只对应一个值

x=-1

。

所以正确答案是 B。

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.

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