1981 AMC 12 Problem 29

Question

If

a\gt1,

then the sum of the real solutions of

\sqrt{a-\sqrt{a+x}}=x

is equal to

\sqrt a-1

\frac{\sqrt a-1}{2}

\sqrt{a-1}

\frac{\sqrt{a-1}}2

\frac{\sqrt{4a-3}-1}{2}

The principal square root forces

x\ge0

After one squaring, factor a difference involving

\sqrt{a+x}-x

Since

x\ge0,

squaring once gives

a-\sqrt{a+x}=x^2.

Let

y=\sqrt{a+x}.

Then

y=a-x^2

and

a=y^2-x,

so

0=(y+x)(y-x-1).

Since

y+x\gt0,

we have

y=x+1,

or

\sqrt{a+x}=x+1.

Therefore

a+x=x^2+2x+1

and

x^2+x+1-a=0.

The only nonnegative root is

x=\frac{\sqrt{4a-3}-1}{2},

and it satisfies the original equation. It is therefore also the sum of all real solutions.

Therefore, the correct answer is E.

Problem 29 in other years