2002 AMC 10B Problem 10

Problem 10 of 25EasierAlgebra

Suppose that

a

and

b

are nonzero real numbers, and that the equation

x^2 + ax + b = 0

has solutions

a

and

b.

What is the pair

(a, b)?

Show solution

Solution

Since the roots are

a

and

b,

Vieta’s formulas give

a + b = -a

and

ab = b.

From

ab = b

with

b \ne 0,

we get

a = 1.

Then

a + b = -a

gives

1 + b = -1,

b = -2.

Thus

(a, b) = (1, -2),

and the correct answer is C.

Concepts: Vieta’s Formulas · system of equations

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.