1967 AMC 12 Problem 17

Attempt Problem 17 of the 1967 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 1967 AMC 12 solutions, or check the answer key.

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17.

r_1

and

r_2

are the distinct real roots of

x^2+px+8=0,

then it must follow that:

\lvert r_1+r_2\rvert\gt4\sqrt2

\lvert r_1\rvert\gt3

\lvert r_2\rvert\gt3

\lvert r_1\rvert\gt2

and

\lvert r_2\rvert\gt2

r_1\lt0

and

r_2\lt0

\lvert r_1+r_2\rvert\lt4\sqrt2

Distinct real roots require the discriminant to be positive

Use

r_1+r_2=-p

The discriminant condition is

p^2-32\gt0,

\lvert p\rvert\gt4\sqrt2.

By Vieta’s formulas,

r_1+r_2=-p,

and therefore

\lvert r_1+r_2\rvert=\lvert p\rvert\gt4\sqrt2.

Therefore, the correct answer is A.

Problem 17 in other years