How to Simplify a Non-Linear Equation: (x - 2)^3 = x^2(x - 6)

[science_rules]

(x - 2)^3 = x^2(x - 6)
= (x^3 - 8) = (x^3 - 6x^2)

(x^3 - 8) = (x^3 - 6x^2)
-x^3 -x^3

= -8 = -6x^2

I am not sure how to solve this for x

 

[Samy_A]

(x - 2)^3 = x^2(x - 6)
= (x^3 - 8) = (x^3 - 6x^2)

(x^3 - 8) = (x^3 - 6x^2)
-x^3 -x^3

= -8 = -6x^2

I am not sure how to solve this for x

(red added by me)

A clue that something went wrong is that $-8=-6x²$ is not a linear equation.
Note that, in general, $(a-b)³ \neq a^³-b^³$, as you seem to think in the expression I set in red.

 

[science_rules]

(x - 2)^3 = x^2(x - 6)
= (x-2)(x-2)(x-2) = x^3 - 6x^2
= (x^2 - 2x -2x + 4) (x-2)
= (x^2 - 4x + 4)(x-2)
= x^3 - 6x^2 + 12x -8
x^3 - 6x^2 + 12x -8 = x^3 -6x^2 ?

 

[science_rules]

I know I am supposed to even out the inequality by subtracting and adding to each side but I am still not sure how to solve for x

 

[Samy_A]

I know I am supposed to even out the inequality by subtracting and adding to each side but I am still not sure how to solve for x

You can start by removing the terms that appear on both sides of the equation. Then see what remains.

 

[science_rules]

x^3 - 6x^2 + 12x -8 = x^3 -6x^2
add 6x^2 to both sides
= x^3 + 12x -8 = x^3
subtract 12x from both sides
= x^3 -8 = x^3 - 12x
subtract x^3 from both sides
= -8 = -12x
x = -8/12
And I believe I got the answer

 

[science_rules]

I meant: x = -8/-12

 

[Samy_A]

I meant: x = -8/-12

Correct. You can simplify this to 8/12 or 2/3.

 

[Mark44]

(x - 2)^3 = x^2(x - 6)
= (x-2)(x-2)(x-2) = x^3 - 6x^2
= (x^2 - 2x -2x + 4) (x-2)
= (x^2 - 4x + 4)(x-2)
= x^3 - 6x^2 + 12x -8
x^3 - 6x^2 + 12x -8 = x^3 -6x^2 ?

Don't connect pairs of equations with '='. An equation is not "equal to" another equation. One equation can be equivalent to another equation (same solution set for both) or one equation can imply another equation. There are separate symbols for each of these relationships, but in your case it would be OK to not use any symbol at all between the successive equations.

I meant: x = -8/-12

You should never leave an answer like this. -8/-12 is the same as 8/12, which can be simplified to 2/3.

Also, this question has nothing to do with linear algebra or abstract algebra, so I moved it to the general math section.

 

[HallsofIvy]

In addition to everything else this is NOT a linear equation. This might have been in a section on "linear equations" because after canceling the "x^3- 6x^2" terms, the remaining equation is linear.