Aim: Solving Complete Cubic Equations
ax^3+bx^2+cx+d = 0

Do Now:
1. If If x^2-10x+21=0 has roots x =3 and x = 7, what are the two roots of (y+5)^2-10(y+5)+ 21=0?

The answer was y -2, and y = 2
2.If (y+3)^2-6(y+3)+5 has roots of y = 2 and y = -2, what are the roots of x^2-6x+5=0?

The answer was x = -5 and x = 1

To solve a full on cubic equation we need to place (x - y) into the equation so that the x^2 term disappears.
(y + k)^3 + b(y + k)^2 + c(y = k) + d = 0

Focusing on the x^3 and x^2 terms, they factor out to y^3 + 3y^2k + 3yk^2 + k^3 and by^2 + b2yk + bk^2 respectively.

We need the y^2 terms gone. So 3y^2k + by^2 = 0 or (3k + b)y^2 = 0
So k = -b/3

So replace x with (y - b/3)

For example
x^3 + 9x^2 + 21x - 31 = 0

becomes

(y - 3)^3 + 9(y - 3)^2 + 21(y - 3) - 31 = 0

Factoring it out, it becomes

y^3 -6y - 40

Then you can use the Depressed Cubic method to get the answer.