1958 AMC 12 Problem 33

Question

For one root of

ax^2+bx+c=0

to be double the other, the coefficients

a,

b,

c

must be related as follows:

4b^2=9c

2b^2=9ac

2b^2=9a

b^2-8ac=0

9b^2=2ac

Represent the two roots as

q

and

2q

Apply Vieta’s formulas to their sum and product, then eliminate

q

Let the roots be

q

and

2q.

Vieta’s formulas give

3q=-\frac ba,\qquad 2q^2=\frac ca.

Squaring the first relation yields

9q^2=\frac{b^2}{a^2}.

Dividing this by the second relation gives

\frac92=\frac{b^2}{ac},

or

2b^2=9ac.

Therefore, the correct answer is B.

Problem 33 in other years