1982 AMC 12 Problem 17

Attempt Problem 17 of the 1982 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 1982 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

17.

How many real numbers

x

satisfy the equation

3^{2x+2}-3^{x+3}-3^x+3=0?

0

1

2

3

4

Substitute

t=3^x,

noting that

t\gt0

The equation becomes a quadratic in

t

Let

t=3^x\gt0.

Then

9t^2-28t+3=0,

whose roots are

t=3

and

t=\frac{1}{9}.

Both are positive and each corresponds to one real

x,

so there are

2

solutions.

Therefore, the correct answer is C.

Problem 17 in other years