1955 AMC 12 Problem 19

Attempt Problem 19 of the 1955 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 1955 AMC 12 solutions, or check the answer key.

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

19.

Two numbers whose sum is

6

and the absolute value of whose difference is

8

are roots of the equation:

x^2-6x+7=0

x^2-6x-7=0

x^2+6x-8=0

x^2-6x+8=0

x^2+6x-7=0

Solve

u+v=6

and

u-v=8

after choosing an order

A monic quadratic with roots

u,v

x^2-(u+v)x+uv

Taking the larger number first,

u+v=6,\qquad u-v=8

gives

u=7

and

v=-1.

Their product is

-7,

so the monic equation with these roots is

x^2-6x-7=0.

Thus, the correct answer is B.

Problem 19 in other years