2013 AMC 10B Problems/Problem 19
Problem
The real numbers form an arithmetic sequence with The quadratic has exactly one root. What is this root?
Solution
Solution 1
It is given that has 1 real root, so the discriminant is zero, or . Because a, b, c are in arithmetic progression, , or . We need to find the unique root, or (discriminant is 0). From , we have . Ignoring the negatives, we have . Fortunately, finding is not very hard. Plug in to , we have , or , and dividing by gives , so . But , violating the assumption that . Therefore, . Plugging this in, we have . But we need the negative of this, so the answer is .
Solution 2
Note that we can divide the polynomial by to make the leading coefficient 1 since dividing does not change the roots or the fact that the coefficients are in an arithmetic sequence. Also, we know that there is exactly one root so this equation is must be of the form where . We now use the fact that the coefficients are in an arithmetic sequence. Note that in any arithmetic sequence, the average is equal to the median. Thus, and . Since , we easily see that has to be between 1 and 0. Thus, we can eliminate and are left with as the answer.
See also
2013 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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