# 1955 AHSME Problems/Problem 32

## Problem 32

If the discriminant of $ax^2+2bx+c=0$ is zero, then another true statement about $a, b$, and $c$ is that:

$\textbf{(A)}\ \text{they form an arithmetic progression}\\ \textbf{(B)}\ \text{they form a geometric progression}\\ \textbf{(C)}\ \text{they are unequal}\\ \textbf{(D)}\ \text{they are all negative numbers}\\ \textbf{(E)}\ \text{only b is negative and a and c are positive}$

## Solution

The discriminant of a quadratic is

$$\Delta = b^2 - 4ac = (2b)^2 - 4ac = 4b^2 - 4ac = 0.$$ We know that $b^2 = ac,$ or $\frac{b}{a} = \frac{c}{b}$ so $a, b, c$ form a geometric progression.

$\boxed{\text{(B)}}$

## See Also

 1955 AHSC (Problems • Answer Key • Resources) Preceded byProblem 31 Followed byProblem 33 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 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 All AHSME Problems and Solutions

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