The discriminant of the equation is
If then the equation has two distinct real roots.
If then the equation has two equal real roots.
If then the equation has no real roots.
2) For x-intercept you have to solve the equation
For y-intercept you have to calculate f(0).
If then the function has a minimum equal to for
If then the function has a maximum equal to for
In this case , so find the maximum.
4) We have
The vertex has the coordinates
Now, solve the system
5) Let and
The maximum of P is the maximum of the function (see 3)