Quadratic equation problem. Composition of functions

Question

Suppose $p(x)$ and $q(x)$ are quadratic polynomials and the three largest roots of $p(q(x))$ are $10$, $20$ and $23$. What is the smallest root of $p(q(x))$?

Then, there will be 4 roots. $q(10)$ $q(20)$ $q(23)$ $q$(another zero) will make the functions equal to zero. The rest...

Answer 1

Hint: Let the roots of $ p(x) = 0 $ be $\alpha, \beta$.

What can you say are the values of $ q(10), q(20), q(23) $?

They are either $ \alpha$ or $\beta$. Note that at most two of them have the same value, since $q(x)$ is a quadratic function.

Hence, can you conclude what the possibilities of the last root is? (Note, there is more than 1 possibility)

We split into casework, depending on what values they are.
Case 1: $q(10) = \alpha, q (20) = \beta , q (23) = \alpha $. The last solution is $r$ and satisfies $ q(r) = \beta$.
Observe that for any constant $C$, the solutions to $q(x) = C$ sum up to the same value by Vieta's formula. Hence, we have $ 10 + 23 = 20 + r$, which gives $r = 13$.

Complete the rest of the cases.