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Difference between revisions of "1951 AHSME Problems/Problem 16"

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Revision as of 20:02, 10 January 2008

Problem

If in applying the quadratic formula to a quadratic equation

\[f(x) \equiv ax^2 + bx + c = 0,\]

it happens that $c = \frac{b^2}{4a}$, then the graph of $y = f(x)$ will certainly:

$\mathrm{(A) \ have\ a\ maximum } \qquad \mathrm{(B) \ have\ a\ minimum} \qquad$ $\mathrm{(C) \ be\ tangent\ to\ the\ x-axis} \qquad$ $\mathrm{(D) \ be\ tangent\ to\ the\ y-axis} \qquad$ $\mathrm{(E) \ lie\ in\ one\ quadrant\ only}$

Solution

The discriminant of the quadratic equation is $b^2 - 4ac = b^2 - 4a\left(\frac{b^2}{4a}\right) = 0$. This indicates that the equation has only one root (applying the quadratic formula, we get $x = \frac{-b + \sqrt{0}}{2a} = -b/2a$). Thus it follows that $f(x)$ touches the x-axis exactly once, and hence is tangent to the x-axis $\Rightarrow \mathrm{(C)}$.

See also

1951 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 15 		Followed by
Problem 17
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions
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