Let x_1 and x_2 be such that x_1≠ x_2 and 3x_i^2-hx_i=b, i=1, 2. Then x_1+x_2 equals

1974 AMC 12 Problem 2

Attempt Problem 2 of the 1974 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 1974 AMC 12 solutions, or check the answer key.

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

Let

x_1

and

x_2

be such that

x_1\ne x_2

and

3x_i^2-hx_i=b,

i=1,

2.

Then

x_1+x_2

equals

-\frac h3

\frac h3

\frac b3

2b

-\frac b3

Both given numbers satisfy the same quadratic equation

Move

b

to the left and use the sum-of-roots formula

The distinct numbers

x_1,x_2

are the two roots of

3x^2-hx-b=0.

By Vieta’s formulas, their sum is

-\frac{(-h)}{3}=\frac{h}{3}.

Therefore, the correct answer is B.