1959 AMC 12 Problem 38

Attempt Problem 38 of the 1959 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1959 AMC 12 solutions, or check the answer key.

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38.

If

4x+\sqrt{2x}=1,

then

x:

is an integer

is fractional

is irrational

is imaginary

may have two different values

Answer: B

Concepts:radical substitution quadratic

Difficulty rating: 1360

Small hint:

Set

u=\sqrt{2x},

so

x=\frac{u^2}{2}

and

u\ge0

Big hint:

Solve the resulting quadratic

2u^2+u-1=0

Solution:

Let

u=\sqrt{2x}\ge0.

Since

x=\frac{u^2}{2},

the equation becomes

2u^2+u=1,

or

(2u-1)(u+1)=0.

Thus

u=\frac{1}{2}

and

x=\frac{u^2}{2}=\frac{1}{8}.

The other quadratic root is inadmissible because

u\ge0.

Hence

x

is fractional.

Therefore, the correct answer is B.

Problem 38 in other years