Difference between revisions of "1965 AHSME Problems/Problem 9"

Revision as of 09:56, 18 July 2024

Problem 9

The vertex of the parabola $y = x^2 - 8x + c$ will be a point on the $x$ -axis if the value of $c$ is:

$\textbf{(A)}\ - 16 \qquad \textbf{(B) }\ - 4 \qquad \textbf{(C) }\ 4 \qquad \textbf{(D) }\ 8 \qquad \textbf{(E) }\ 16$

Solution

Notice that if the vertex of a parabola is on the x-axis, then the x-coordinate of the vertex must be a solution to the quadratic. Since the quadratic is strictly increasing on either side of the vertex, the solution must have double multiplicity, or the quadratic is a perfect square trinomial. This means that for the vertex of $y = x^2 - 8x + c$ to be on the x-axis, the trinomial must be a perfect square, and have discriminant of zero. So,

$\begin{align*} 0 &= b^2-4ac\\ 0 &= (-8)^2-4c\\ c &= 64\\ c &= 16\\ \end{align*}$

Therefore $c=16$ , and our answer is $\boxed{\textbf{(E)}}$ .

1965 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
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 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40
All AHSME Problems and Solutions

@@ Line 20: / Line 20: @@
 \end{align*}</cmath>
-Therefore <math>c=16</math>, and our answer is <math>\boxed{\textbf{(E)}}</math>
+Therefore <math>c=16</math>, and our answer is <math>\boxed{\textbf{(E)}}</math>.
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Difference between revisions of "1965 AHSME Problems/Problem 9"

Revision as of 09:56, 18 July 2024

Problem 9

Solution

See Also