Is (a+b)^2 Always Equal to a^2+2ab+b^2?

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Homework Help Overview

The discussion centers around the algebraic expression (a+b)^2 and its equivalence to a^2+2ab+b^2. Participants explore whether this equality holds universally, particularly in different mathematical contexts such as commutative rings and non-commutative structures.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Some participants attempt to clarify the conditions under which the equality holds, noting that it is valid in the context of real numbers but may not apply in other mathematical structures like matrices. Others question the implications of the equality in different scenarios.

Discussion Status

The discussion is ongoing, with participants providing insights into the conditions for the equality to hold. There is exploration of different interpretations regarding the nature of the variables involved, but no consensus has been reached.

Contextual Notes

Participants reference the context of high school mathematics and the implications of algebraic rules in various mathematical settings, indicating a focus on foundational understanding.

Wardlaw
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Math problem!

Homework Statement



Is (a+b)^2 ALWAYS equal to a^2+2ab+b^2?

Homework Equations





The Attempt at a Solution

 
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Wardlaw said:

Homework Statement



Is (a+b)^2 ALWAYS equal to a^2+2ab+b^2?

Homework Equations


The Attempt at a Solution


Yes, if a and b represent objects in a commutative ring like the real numbers. If multiplication is not commutative, (for example a and b are matrices) then (a+b)^2=a^2+ab+ba+b^2 which is not generally the same. Why do you ask?
 
Last edited:


Wardlaw said:

Homework Statement



Is (a+b)^2 ALWAYS equal to a^2+2ab+b^2?

Homework Equations





The Attempt at a Solution


If a and b are numbers as they should be if you're in high school, then yes, they're always equal. Why?

[tex](a+b)^2[/tex]
[tex]=(a+b)(a+b)[/tex]
[tex]=a(a+b)+b(a+b)[/tex]

And I'm sure you can expand out the rest.
 


Math formulas generally don't have expiration or 'black out' dates.
 


Well this was weird.. xD
 


Is p+q+r=p+q+s never valid?
 


Wardlaw said:
Is p+q+r=p+q+s never valid?

Only if r=s. Why do you want to know?
 

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