10.1. It is known that the quadratic equations in $x$, $2017 x^{2} + p x + q = 0$ and $u p x^{2} + q x + 2017 = 0$ (where $p$ and $q$ are given real numbers) have one common root. Find all possible values of this common root and prove that there are no others.
Solution: Let $x_{0}$ be the common root of these equations, that is, the equalities $2017 x_{0}^{2} + p x_{0} + q = 0$ and $p x_{0}^{2} + q x_{0} + 2017 = 0$ are satisfied. Multiply both sides of the first equation by the number $x_{0}$ and subtract the second equation. We get the consequential equation: $2017 x_{0}^{3} - 2017 = 0$, from which $x_{0} = 1$. Substituting this root into any of the original equations, we get the equality $p + q + 2017 = 0$. This means that when $p + q = -2017$, both equations indeed have a common root equal to 1, and if this sum equals another number, there are simply no common roots.

Answer: 1.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| Correct and fully justified answer | 7 points |
| Correctly written roots of both quadratic equations and at least one case of coinciding roots is correctly investigated | 4 points |
| Noted that if $1$ is a root, then this root is common, but not proven that there can be no other common roots | 1 point |
| Correct answer without justification and/or calculations from which the solution process is not visible | 0 points |