Proving a^2 - b^2 = (a+b)(a-b) with Factorization

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Discussion Overview

The discussion revolves around proving the identity \( a^2 - b^2 = (a+b)(a-b) \) using factorization techniques. Participants explore various algebraic manipulations and approaches to arrive at the factorization.

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant asks how to prove the identity using factorization and suggests starting with \( a*a - b*b \).
  • Another participant attempts to manipulate the expression by adding and subtracting \( ab \) to derive a different form of \( a^2 - b^2 \).
  • There is a question about how to transition from \( a^2 - b^2 \) to \( a^2 - b^2 + ab - ab \).
  • One participant suggests that adding and subtracting \( ab \) is a valid step, emphasizing that this is equivalent to adding zero.
  • Another participant reiterates the method of adding and subtracting \( ab \) but seeks clarification on the initial transformation.

Areas of Agreement / Disagreement

The discussion does not reach a consensus on the steps required to prove the identity, with participants expressing uncertainty about the algebraic manipulations involved.

Contextual Notes

Participants express confusion regarding the justification for adding and subtracting \( ab \) and the overall approach to factorization, indicating potential gaps in understanding the algebraic principles at play.

scientifico
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Hello, how can I prove using factorization that a^2 - b^2 = (a+b)(a-b) ?

a*a - b*b I should get something like a(a-b)+b(a-b) but how ?

thanks
 
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a2 - b2 == a2 - b2 + ab - ab == a2 + ab - b2 - ab == a(a+b) - b(a+b) == ?
 
Ok but how do I get a^2 - b^2 + ab - ab from a^2 - b^2 ?
 
Add ab and subtract ab.
 
scientifico said:
Ok but how do I get a^2 - b^2 + ab - ab from a^2 - b^2 ?

HallsofIvy said:
Add ab and subtract ab.

You can always add zero to any expression (here in the form of ab + (-ab), and you can always multiply an expression by 1.
 

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