# Problem 1. (2 points)

Two given quadratic trinomials differ by the permutation of the free term and the second coefficient. The sum of these trinomials has a single root. What value does this sum take at one?
Answer: 2 or 18.

## Solution:

Let these quadratic polynomials be of the form $x^{2}+p x+q$ and $x^{2}+q x+p$. Then their sum is $2 x^{2}+(p+q) x+(p+q)$. The discriminant of this quadratic polynomial is $(p+q)^{2}-8(p+q)$, from which $p+q=0$ or $p+q=8$. In the first case, the sum is 2, and in the second case, 18.

It is clear that both values are achievable.