1957 AMC 12 Problem 2

Attempt Problem 2 of the 1957 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 1957 AMC 12 solutions, or check the answer key.

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

In the equation

2x^2-hx+2k=0,

the sum of the roots is

4

and the product of the roots is

-3.

Then

h

and

k

have the values, respectively:

8

and

-6

4

and

-3

-3

and

4

-3

and

8

8

and

-3

Apply Vieta’s formulas without first solving the quadratic

The sum is

\frac{h}{2}

and the product is

k

By Vieta’s formulas, the sum and product of the roots are

\begin{aligned} \frac h2&=4,\\ \frac{2k}{2}&=k=-3. \end{aligned}

Hence

h=8

and

k=-3.

Therefore, the correct answer is E.

Problem 2 in other years