1956 AMC 12 Problem 7

Attempt Problem 7 of the 1956 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 1956 AMC 12 solutions, or check the answer key.

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The roots of the equation

ax^2+bx+c=0

will be reciprocal if:

a=b

a=bc

c=a

c=b

c=ab

If the roots are

r

and

s,

reciprocal roots satisfy

rs=1

Use Vieta’s formula

rs=\frac{c}{a}

By Vieta’s formulas, the product of the two roots is

\frac{c}{a}.

Reciprocal roots have product

1,

\frac ca=1,

which requires

c=a.

Thus, the correct answer is C.

Problem 7 in other years