The first step is to simplify both sides of the equation. The equation can be written as
y^3 - 3^3 = 9y(y - 3)
For the left hand side, we would apply the difference of two cubes formula. it is expressed as
x^3 - y^3 = (x - y)(x^2 + xy + y^2)
By comparing with the left hand side of the equation,
x = y and y = 3. It becomes
(y - 3)(y^2 + 3y + 3^2)
= (y - 3)(y^2 + 3y + 9)
The equation becomes
(y - 3)(y^2 + 3y + 9) = 9y(y - 3)
If we divide both sides of the equation by (y - 3), it becomes
(y - 3)(y^2 + 3y + 9)/(y - 3 = 9y(y - 3)/(y - 3)
y^2 + 3y + 9 = 9y
y^2 + 3y - 9y + 9 = 0
y^2 - 6y + 9 = 0
We would solve the quadratic equation by applying the method of factorisation. We would find two terms such that their sum or difference is - 6y and their product is 9y^2. The terms are - 3y and - 3y. The equation becomes
y^2 - 3y - 3y + 9
y(y - 3) - 3( y - 3) = 0
(y - 3)(y - 3) = 0
y - 3 = 0 twice
y = 3 twice