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

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## Revision as of 21:46, 3 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

Alternatively, note that whatever the two values of are, they must lead to equations of the form and . So the two choices of must make and so and .