1958 AMC 12 第 11 题

Question

满足方程

\sqrt{5-x}=x\sqrt{5-x}

的根的个数为：

The number of roots satisfying the equation

\sqrt{5-x}=x\sqrt{5-x}

is:

无限多个

unlimited

3

2

1

0

将两边移到同一边，并提取公因式根式

Move both sides to one side and factor the common radical

分别检查根式为零和另一个因式为零的情形

Check separately when the radical is zero and when its remaining factor is zero

因式分解得

(1-x)\sqrt{5-x}=0\text{。}

因此

x=1

，或由

5-x=0

得

x=5

。两个值都在定义域

x\le5

内，并且满足原方程。

所以共有两个根，正确答案为 C。

Factoring gives

(1-x)\sqrt{5-x}=0.

Thus either

x=1

or

5-x=0,

giving

x=5.

Both values lie in the domain

x\le5

and satisfy the original equation.

Therefore, there are two roots, and the correct answer is C.

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