# 1964 AHSME Problems/Problem 7

## Problem

Let n be the number of real values of $p$ for which the roots of $x^2-px+p=0$ are equal. Then n equals: $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \text{a finite number greater than 2}\qquad \textbf{(E)}\ \infty$

## Solution

If the roots of the quadratic $Ax^2 + Bx + C = 0$ are equal, then $B^2 - 4AC = 0$. Plugging in $A=1, B=-p, C = p$ into the equation gives $p^2 - 4p = 0$. This leads to $p = 0, 4$, so there are two values of $p$ that work, giving answer $\boxed{\textbf{(C)}}$.

## See Also

 1964 AHSC (Problems • Answer Key • Resources) Preceded byProblem 6 Followed byProblem 8 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

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