2021 Fall AMC 12A Problem 23

A quadratic polynomial with real coefficients and leading coefficient 1 is called \textit{disrespectful} if the equation p(p(x))=0 is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial \tilde{p}(x) for which the sum of the roots is maximized. What is \tilde{p}(1)?

\textbf{(A) } \frac{5}{16} \qquad\textbf{(B) } \frac{1}{2} \qquad\textbf{(C) } \frac{5}{8} \qquad\textbf{(D) } 1 \qquad\textbf{(E) } \frac{9}{8}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.