Question

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

they form an arithmetic progression

they form a geometric progression

they are unequal

they are all negative numbers

only $b$ is negative and $a$ and $c$ are positive

Answer: B

Concepts:quadratic discriminant geometric progression

Difficulty rating: 1260

Small Hint:

Set $(2b)^2-4ac$ equal to zero

Big Hint:

Compare the resulting relation with the defining tests for arithmetic and geometric progressions

Solution:

A zero discriminant gives $(2b)^2-4ac=0,$ hence $b^2=ac.$ Equivalently, $\frac{a}{b}=\frac{b}{c}$ where the ratios are defined, which is the defining relation for $a,b,c$ to form a geometric progression.

Thus, the correct answer is B.

1955 AMC 12 Problem 32

Answer: B

Difficulty rating: 1260

Small Hint:

Big Hint:

Solution:

Problem 32 in Other Years