2011 AMC 10B 第 19 题

Question

下列方程所有根的乘积是多少？

\sqrt{5 | x | + 8} = \sqrt{x^2 - 16}

What is the product of all the roots of the equation below?

\sqrt{5 | x | + 8} = \sqrt{x^2 - 16}

-64

-24

-9

24

576

检查根号有定义后，将两边平方。

Square both sides after checking the radicals are defined

用

|x|

写出方程。

Write the equation in terms of

|x|

因为

x^2=|x|^2

，将方程两边平方可得

\begin{aligned} 5|x|+8 &= |x|^2-16,\\ |x|^2-5|x|-24&=0,\\ (|x|-8)(|x|+3)&=0 \end{aligned}\text{。}

|x|

在代数上可能为

8

或

-3

，但绝对值不能为负。因此

|x|=8

，两个根为

8

和

-8

，乘积为

-64

。

所以正确答案是 A。

Because

x^2=|x|^2,

squaring the equation gives

\begin{aligned} 5|x|+8 &= |x|^2-16,\\ |x|^2-5|x|-24&=0,\\ (|x|-8)(|x|+3)&=0. \end{aligned}

The algebraic possibilities for

|x|

are

8

and

-3,

but an absolute value cannot be negative. Hence

|x|=8,

so the roots are

8

and

-8

, with product

-64.

Thus, the correct answer is A .

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