1955 AMC 12 Problem 20

Question

The expression

\sqrt{25-t^2}+5

equals zero for:

no real or imaginary values of

t

no real values of

t

only

no imaginary values of

t

only

t=0

t=\pm5

Answer: A

Concepts:principal square root radical equation complex numbers

Difficulty rating: 1340

Small hint:

Isolate the radical before doing any algebra

Big hint:

If you square, check every candidate in the original equation because the radical has a prescribed sign

Solution:

For the expression to vanish, one would need

\sqrt{25-t^2}=-5.

The radical sign denotes the principal square root. It is nonnegative for a nonnegative real radicand and, over the complex numbers, is chosen with nonnegative real part; it therefore cannot equal

-5.

Squaring would introduce the extraneous candidate

t=0,

for which the original expression is

10.

Thus, no real or imaginary value works, and the correct answer is A.