1965 AMC 12 Problem 7

Attempt Problem 7 of the 1965 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 1965 AMC 12 solutions, or check the answer key.

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

The sum of the reciprocals of the roots of the equation

ax^2+bx+c=0

is:

\dfrac1a+\dfrac1b

-\dfrac cb

\dfrac bc

-\dfrac ab

-\dfrac bc

If the roots are

r,s,

then

\frac{1}{r}+\frac{1}{s}=\frac{r+s}{rs}

Use

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

and

rs=\frac{c}{a}

By Vieta’s formulas,

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

and

rs=\frac{c}{a}.

Therefore

\begin{aligned} \frac1r+\frac1s &=\frac{r+s}{rs}\\ &=\frac{-\frac{b}{a}}{\frac{c}{a}}=-\frac bc. \end{aligned}

Thus, the correct answer is E.

Problem 7 in other years