# Difference between revisions of "2005 AMC 10A Problems/Problem 10"

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==Solution== | ==Solution== | ||

A [[quadratic equation]] has exactly one [[root]] if and only if it is a [[perfect square]]. So set | A [[quadratic equation]] has exactly one [[root]] if and only if it is a [[perfect square]]. So set | ||

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<math>4x^2 + ax + 8x + 9 = (mx + n)^2</math> | <math>4x^2 + ax + 8x + 9 = (mx + n)^2</math> | ||

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<math>4x^2 + ax + 8x + 9 = m^2x^2 + 2mnx + n^2</math> | <math>4x^2 + ax + 8x + 9 = m^2x^2 + 2mnx + n^2</math> | ||

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Two [[polynomial]]s are equal only if their [[coefficient]]s are equal, so we must have | Two [[polynomial]]s are equal only if their [[coefficient]]s are equal, so we must have | ||

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<math>m^2 = 4, n^2 = 9</math> | <math>m^2 = 4, n^2 = 9</math> | ||

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<math>m = \pm 2, n = \pm 3</math> | <math>m = \pm 2, n = \pm 3</math> | ||

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<math>a + 8= 2mn = \pm 2\cdot 2\cdot 3 = \pm 12</math> | <math>a + 8= 2mn = \pm 2\cdot 2\cdot 3 = \pm 12</math> | ||

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<math>a = 4</math> or <math>a = -20</math>. | <math>a = 4</math> or <math>a = -20</math>. | ||

So the desired sum is <math> (4)+(-20)=-16 \Longrightarrow \mathrm{(A)} </math> | So the desired sum is <math> (4)+(-20)=-16 \Longrightarrow \mathrm{(A)} </math> | ||

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==See Also== | ==See Also== | ||

*[[2005 AMC 10A Problems]] | *[[2005 AMC 10A Problems]] |

## Revision as of 10:49, 2 August 2006

## Problem

There are two values of for which the equation has only one solution for . What is the sum of those values of ?

## Solution

A quadratic equation has exactly one root if and only if it is a perfect square. So set

Two polynomials are equal only if their coefficients are equal, so we must have

or .

So the desired sum is