Challenge Level

Start with the linear polynomial: $y = -3x + 9$. The
$x$-coefficient, the root and the intercept are -3, 3 and 9
respectively, and these are in arithmetic progression. Are there
any other linear polynomials that enjoy this property?

What about quadratic polynomials? That is, if the polynomial \[y = ax^2 + bx + c\] has roots $r_1$ and $r_2,$ can $a$, $r_1$, $b$, $r_2$ and $c$ be in arithmetic progression?

[The idea for this problem came from Janusz Kowalski of the Kreator Project.]

What about quadratic polynomials? That is, if the polynomial \[y = ax^2 + bx + c\] has roots $r_1$ and $r_2,$ can $a$, $r_1$, $b$, $r_2$ and $c$ be in arithmetic progression?

[The idea for this problem came from Janusz Kowalski of the Kreator Project.]