# 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 |

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

<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> |

## Revision as of 09:49, 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