The number of roots satisfying the equation √5-x=x√5-x is:

1958 AMC 12 Problem 11

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11.

The number of roots satisfying the equation

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

is:

unlimited

3

2

1

0

Move both sides to one side and factor the common radical

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

Factoring gives

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

Thus either

x=1

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.