2005 AMC 12A Problems/Problem 9
Contents
Problem
There are two values of for which the equation has only one solution for . What is the sum of these values of ?
Solution
Solution 1
A quadratic equation always has two roots, unless it has a double root. That means we can write the quadratic as a square, and the coefficients 4 and 9 suggest this. Completing the square, , so . The sum of these is .
Solution 2
Another method would be to use the quadratic formula, since our coefficient is given as 4, the coefficient is and the constant term is . Hence, Because we want only a single solution for , the determinant must equal 0. Therefore, we can write which factors to ; using Vieta's formulas we see that the sum of the solutions for is the opposite of the coefficient of , or .
Solution 3
Using the discriminant, the result must equal . Therefore, or , giving a sum of .
Solution 4
First, notice that for there to be only root to a quadratic, the quadratic must be a square. Then, notice that the quadratic and linear terms are both squares. Thus, the value of must be such that both and . Clearly, or . Hence .
Solution by franzliszt
See also
2005 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 8 |
Followed by Problem 10 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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