1) Let

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).

3) Let

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

Then

and

Now, solve the system

5) Let and

Then

The maximum of P is the maximum of the function (see 3)