1978 AMC 12 Problem 13

Attempt Problem 13 of the 1978 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 1978 AMC 12 solutions, or check the answer key.

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13.

a,

b,

c,

and

d

are nonzero numbers such that

c

and

d

are the solutions of

x^2+ax+b=0

and

a

and

b

are the solutions of

x^2+cx+d=0,

then

a+b+c+d

equals

0

-2

2

4

\frac{-1+\sqrt5}{2}

Apply Vieta’s formulas to both quadratics

The two sum equations imply

b=d

; then use the product equations and nonzero condition

Vieta’s formulas give

\begin{aligned} c+d&=-a,& cd&=b,\\ a+b&=-c,& ab&=d. \end{aligned}

The two sum equations imply

b=d.

Since

b=d\ne0,

the product equations give

a=c=1.

Then

a+b=-c

gives

b=d=-2,

and therefore

a+b+c+d=-2.

Therefore, the correct answer is B.

Problem 13 in other years