2019 AMC 12B Problems/Problem 21
Problem
How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is 
and the roots are 
and 
then the requirement is that 
.)
Solution
Because there are three coefficients and two roots, we need at least two elements in the set 
to be equal to each other. It is possible that all three could be equal to each other. In the case that two elements in the set are equal to each other, two of those elements will be equal to and the third will be equal to 
.
Case 1: 
We would need the polynomial 
to have a double root . By inspection, there is no such polynomial, so there are no polynomials for this case.
Case 2: 
and 
The polynomial will be in the form 
. By Vieta's formulas, 
and 
. The second equation tells us that either 
or 
. Testing each possibility, we find the polynomials 
and 
, both of which work. There are 2 polynomials for this case.
Case 3: 
and 
The polynomial will be in the form 
. By Vieta's formulas, 
and 
. Through substitution, we get 
. The function f(r) = is a strictly increase function with one real root.
[Work in Progress]
See Also
| 2019 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 20 |
Followed by Problem 22 |
| 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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