1955 AMC 12 Problem 32

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

Set

(2b)^2-4ac

equal to zero

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

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.

Problem 32 in other years