2008 AMC 10B Problem 9

Attempt Problem 9 of the 2008 AMC 10B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2008 AMC 10B solutions, or check the answer key.

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A quadratic equation

ax^2-2ax+b=0

has two real solutions. What is the average of the solutions?

1

2

\dfrac{b}{a}

\dfrac{2b}{a}

\sqrt{2a-b}

For

Ax^2+Bx+C=0,

the sum of the roots is

-\dfrac{B}{A}

The average is half the sum of the roots

By Vieta’s formulas, the sum of the roots of

ax^2-2ax+b=0

\dfrac{-(-2a)}{a}=2.

The average is half of this, namely

1.

Thus, the correct answer is A.

Problem 9 in other years