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

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Discussion Overview

The discussion revolves around the simplification and solution of the non-linear equation (x - 2)^3 = x^2(x - 6). Participants explore various approaches to manipulate the equation and express uncertainty about the correct method to solve for x.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants express confusion about the manipulation of the equation, particularly regarding the transition from (x - 2)^3 to x^3 - 6x^2.
  • One participant notes that the equation -8 = -6x^2 is not linear, suggesting a misunderstanding in the simplification process.
  • Another participant attempts to clarify the expansion of (x - 2)^3 and its equivalence to x^3 - 6x^2, but questions arise about the correctness of the steps taken.
  • Several participants mention the need to balance the equation by adding or subtracting terms, but express uncertainty about how to proceed effectively.
  • A later reply suggests removing common terms from both sides of the equation as a potential strategy for simplification.
  • One participant arrives at a solution of x = -8/-12, which is later simplified to 2/3, but there is a discussion about the appropriateness of presenting the answer in its unsimplified form.
  • There is a correction regarding the nature of the equation, with some participants emphasizing that it is not a linear equation despite the appearance of a linear form after certain terms are canceled.

Areas of Agreement / Disagreement

Participants generally do not reach a consensus on the best method to simplify and solve the equation. There are multiple competing views on the steps involved and the nature of the equation itself.

Contextual Notes

Some participants highlight limitations in their understanding of the algebraic manipulations involved, and there is an acknowledgment of the potential confusion arising from the equation's non-linear characteristics.

science_rules
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(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
 
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science_rules said:
(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.
 
Last edited:
(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 ?
 
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
 
science_rules said:
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.
 
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
 
I meant: x = -8/-12
 
science_rules said:
I meant: x = -8/-12
Correct. You can simplify this to 8/12 or 2/3.
 
science_rules said:
(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.
science_rules said:
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.
 
  • #10
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 "[itex]x^3- 6x^2[/itex]" terms, the remaining equation is linear.
 

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