1962 AMC 12 Problem 34

Question

For what real values of

K

does

x=K^2(x-1)(x-2)

have real roots?

none

-2\lt K\lt1

-2\sqrt2\lt K\lt2\sqrt2

K\gt1

or

K\lt-2

all

Expand and collect terms to obtain a quadratic in

x

Its discriminant simplifies to

K^4+6K^2+1

Rearranging gives

K^2x^2-(3K^2+1)x+2K^2=0.

Its discriminant is

\begin{aligned} &(3K^2+1)^2-8K^4\\ &\qquad=K^4+6K^2+1\gt0 \end{aligned}

for every real

K.

This also covers

K=0,

when the original equation gives

x=0.

Therefore, the correct answer is E.

Problem 34 in other years