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:

$4b^2=9c$

$2b^2=9ac$

$2b^2=9a$

$b^2-8ac=0$

$9b^2=2ac$

Answer: B

Concepts:quadratic Vieta’s Formulas algebraic manipulation

Difficulty rating: 1630

Small Hint:

Represent the two roots as $q$ and $2q$

Big Hint:

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

Solution:

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.

1958 AMC 12 Problem 33

Answer: B

Difficulty rating: 1630

Small Hint:

Big Hint:

Solution:

Problem 33 in Other Years