2008 AMC 10B Problems/Problem 9

Problem

A quadratic equation $ax^2 - 2ax + b = 0$ has two real solutions. What is the average of these two solutions?

$\mathrm{(A)}\ 1\qquad\mathrm{(B)}\ 2\qquad\mathrm{(C)}\ \frac ba\qquad\mathrm{(D)}\ \frac{2b}a\qquad\mathrm{(E)}\ \sqrt{2b-a}$

Solution 1

Dividing both sides by $a$ , we get $x^2 - 2x + b/a = 0$ . By Vieta's formulas, the sum of the roots is $2$ , therefore their average is $1\Rightarrow \boxed{A}$ .

Solution 2

We know that for an equation $ax^2 + bx + c = 0$ , the sum of the roots is $-b/a$ . This means that the sum of the roots for $ax^2 - 2ax + b = 0$ is $\frac{2a}{a}$ , or $2$ . The average is the sum of the two roots divided by two, so the average is $\frac22 = 1$ .

2008 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
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2008 AMC 10B Problems/Problem 9

Contents

Problem

Solution 1

Solution 2

See also