The equation x+√x-2=4 has:

(B) 1 real and 1 imaginary root

1950 AMC 12 Problem 24

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24.

The equation

x+\sqrt{x-2}=4

has:

2

real roots

1

real and

1

imaginary root

2

imaginary roots

No roots

1

real root

Isolate the square root before squaring

Check every root of the resulting quadratic in the original equation

Isolating and squaring gives

\begin{aligned} \sqrt{x-2}&=4-x,\\ x-2&=(4-x)^2,\\ x^2-9x+18&=0. \end{aligned}

The quadratic roots are

3

and

6.

The value

3

satisfies the original equation, but

6

gives

6+\sqrt4=8,

so it is extraneous.

Thus, the correct answer is E.