1951 AMC 12 Problem 16

Attempt Problem 16 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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16.

If in applying the quadratic formula to a quadratic equation

f(x)=ax^2+bx+c=0,

it happens that

c=\dfrac{b^2}{4a},

then the graph of

y=f(x)

will certainly:

Have a maximum

Have a minimum

Be tangent to the

x

-axis

Be tangent to the

y

-axis

Lie in one quadrant only

Substitute the condition into the discriminant

b^2-4ac

A quadratic with one repeated real root touches the horizontal axis once

The condition gives

b^2-4ac=b^2-4a\left(\frac{b^2}{4a}\right)=0.

Hence the quadratic has a repeated real root, so its parabola is tangent to the

x

-axis.

Thus, the correct answer is C.

Problem 16 in other years