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

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The expression (a+b)^2 is always equal to a^2 + 2ab + b^2 when a and b are elements of a commutative ring, such as real numbers. In cases where multiplication is not commutative, like with matrices, the equality does not hold, as (a+b)^2 expands to a^2 + ab + ba + b^2. For high school mathematics, the equality is consistently valid. The discussion also briefly touches on the validity of another equation, p + q + r = p + q + s, which holds true only if r equals s. Overall, the mathematical principles governing these equations remain consistent and reliable.
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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?

(a+b)^2
=(a+b)(a+b)
=a(a+b)+b(a+b)

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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