2002 AMC 10B Problem 10

Attempt Problem 10 of the 2002 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 2002 AMC 10B solutions, or check the answer key.

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10.

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)?

(-2, 1)

(-1, 2)

(1, -2)

(2, -1)

(4, 4)

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

-a

and the product is

b

a + b = -a

and

ab = b

; use

b \ne 0

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.

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