1951 AMC 12 Problem 47

Attempt Problem 47 of the 1951 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 1951 AMC 12 solutions, or check the answer key.

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47.

If

r

and

s

are the roots of the equation

ax^2+bx+c=0,

the value of

\dfrac1{r^2}+\dfrac1{s^2}

is:

b^2-4ac

\dfrac{b^2-4ac}{2a}

\dfrac{b^2-4ac}{c^2}

\dfrac{b^2-2ac}{c^2}

None of these

Answer: D

Concepts:Vieta’s Formulas algebraic manipulation

Difficulty rating: 1530

Small hint:

Write the expression over the common denominator

r^2s^2

Big hint:

Use

r+s=-\frac{b}{a}

and

rs=\frac{c}{a}

Solution:

By Vieta’s formulas,

r+s=-\frac ba,\qquad rs=\frac ca.

Therefore

\begin{aligned} \frac1{r^2}+\frac1{s^2} &=\frac{(r+s)^2-2rs}{(rs)^2}\\ &=\frac{b^2-2ac}{c^2}. \end{aligned}

Thus, the correct answer is D.

Problem 47 in other years