Say we have:
Divide by a , to rewrite this as:
Put y=x+f/4 , so x=y−f/4 , so
=y4+y3(−f+f) y2(3f2/8)−32/4+4) +y(−f3/16+3f3/16−gf/2+h) +
with p=−3f2/8+g ,
Now, we can write:
Now, for any z ,
The right hand side of (*) is a quadratic in y ; and we can choose z so that it is a perfect square, i.e. so that the discriminant is zero, ie:
We can rewrite this as (q2−4p3+4pr)+(−16p2+8r)z−20pz2−8z3=0 .
This is a cubic equation in z . So we can solve it for z using the cubic equation formula (Cardano's method).
When we have solved this to find a value for z , we can substitute in this value of z to (*). This makes the right hand side of (*) a perfect square, so we can take the square root of both sides of (*). (*) is then a quadratic equation in y , which we can solve using the quadratic solution formula. The values of y will easily give us values of x , i.e. solutions of the original equation.
There can be 0, 1, 2, 3, or 4 real solutions.