1972 AMC 12 Problem 22

Attempt Problem 22 of the 1972 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 1972 AMC 12 solutions, or check the answer key.

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22.

a\pm bi

(b\ne0)

are imaginary roots of the equation

x^3+qx+r=0,

where

a,

b,

q,

and

r

are real numbers, then

q

in terms of

a

and

b

is:

a^2+b^2

2a^2-b^2

b^2-a^2

b^2-2a^2

b^2-3a^2

The conjugate

a-bi

is another root

Because the

x^2

coefficient is zero, determine the third root and then use the sum of pairwise products

The conjugate root is

a-bi.

Since the sum of the three roots is

0,

the third root is

-2a.

By Vieta’s formulas,

\begin{aligned} q&=(a+bi)(a-bi)\\ &\quad-2a(a+bi)\\ &\quad-2a(a-bi)\\ &=a^2+b^2-4a^2\\ &=b^2-3a^2. \end{aligned}

Therefore, the correct answer is E.

Problem 22 in other years