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Solving a third degree equation

Solve the equation

x^3 - 3x^2 + 3 = 0

Solution Preview

To solve an equation of the form:

x^3 + a x^2 + b x + c = 0

follow the following steps.

First, get rid of the quadratic term a x^2 using the substitution:

x = y - a/3

This step is analogous to how you solve the quadratic equation when you write it as a perfect square.

In this case we aren't finshed yet as the equation now is of the form:

y^3 + p y + q = 0

How do we solve this equation?

Consider the identity:

(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

We can rewrite the right hand side as follows:

a^3 + 3a^2b + 3ab^2 + b^3 =

a^3 + b^3 + 3ab(a+b)

So, we have:

(a+b)^3 = 3ab(a+b)+ a^3 + b^3 ---->

(a+b)^3 - 3ab(a+b) -([a^3+b^3) = 0

Of course, this equation is always ...

Solution Summary

I show how to solve general third degree equations. I work out the details explicitely for the equation x^3 - 3x^2 + 3 = 0

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