If r and s are the roots of x^2-px+q=0, then r^2+s^2 equals:

1955 AMC 12 Problem 27

Attempt Problem 27 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.

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27.

r

and

s

are the roots of

x^2-px+q=0,

then

r^2+s^2

equals:

p^2+2q

p^2-2q

p^2+q^2

p^2-q^2

p^2

Use

r+s=p

and

rs=q

Expand

(r+s)^2

and isolate

r^2+s^2

By Vieta’s formulas,

r+s=p

and

rs=q.

Therefore

\begin{aligned} r^2+s^2 &=(r+s)^2-2rs\\ &=p^2-2q. \end{aligned}

Thus, the correct answer is B.