1954 AMC 12 Problem 24

Attempt Problem 24 of the 1954 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 1954 AMC 12 solutions, or check the answer key.

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

24.

The values of

k

for which the equation

2x^2-kx+x+8=0

will have real and equal roots are:

9

and

-7

only

-7

9

and

7

-9

and

-7

only

9

Combine the linear terms to write the coefficient of

x

1-k

Set the discriminant

(1-k)^2-64

equal to zero

A repeated real root requires

(1-k)^2-4(2)(8)=0.

Thus

(1-k)^2=64,

1-k=\pm8.

The two values are

k=9

and

k=-7.

Thus, the correct answer is A.

Problem 24 in other years