Can I Factor 4a^2b^2 - 9(ab + c)^2?

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Discussion Overview

The discussion revolves around the factorization of the expression 4a^2b^2 - 9(ab + c)^2. Participants explore different approaches to factor the expression, including the use of the difference of squares and the implications of rewriting terms.

Discussion Character

  • Mathematical reasoning, Debate/contested

Main Points Raised

  • One participant suggests using the FOIL method on (ab + c)^2 as a first step in the factorization process.
  • Another participant proposes rewriting the expression as $$(2ab)^2-(3(ab+c))^2$$ to facilitate factoring as a difference of squares.
  • A subsequent reply questions the introduction of the factor of 3, seeking clarification on its origin and its relation to the expression.
  • Another participant reiterates the reasoning behind rewriting 9(ab + c)^2 as (3(ab + c))^2, emphasizing the mathematical validity of this transformation.
  • One participant indicates they will return to complete the problem later, suggesting ongoing engagement with the topic.

Areas of Agreement / Disagreement

Participants express differing views on the factorization process, particularly regarding the treatment of the term 9(ab + c)^2 and the introduction of the factor of 3. The discussion remains unresolved as participants clarify their perspectives without reaching consensus.

Contextual Notes

There are unresolved questions regarding the steps taken in the factorization process, particularly the assumptions about the terms involved and the implications of rewriting expressions.

mathdad
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Factor the expression.

4a^2b^2 - 9(ab + c)^2

Should I FOIL (ab + c)^2 as step 1?

I can then distribute 9 as step 2, right?
 
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I would first write the expression as:

$$(2ab)^2-(3(ab+c))^2$$

Now, factor as the difference of squares, and simplify. :D
 
MarkFL said:
I would first write the expression as:

$$(2ab)^2-(3(ab+c))^2$$

Now, factor as the difference of squares, and simplify. :D

The number 9 is not part of the squaring for (ab + c)^2. So, where did 3 come from? Does 3 come from the fact that 3^2 is 9? This makes sense to me if 9 were inside the parentheses for (ab + c)^2 but it's not. See what I mean?
 
RTCNTC said:
The number 9 is not part of the squaring for (ab + c)^2. So, where did 3 come from? Does 3 come from the fact that 3^2 is 9? This makes sense to me if 9 were inside the parentheses for (ab + c)^2 but it's not. See what I mean?

I was just rewriting that term as a perfect square:

$$9(ab+c)^2=3^2(ab+c)^2=(3(ab+c))^2$$
 
MarkFL said:
I was just rewriting that term as a perfect square:

$$9(ab+c)^2=3^2(ab+c)^2=(3(ab+c))^2$$

Ok. I will complete this problem tomorrow.
 

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