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Factoring

by HeyArnold » Sun Aug 07, 2011 8:04 am
Given that [A, B] = A^2 + B^2 + 2AB, what is A + B if [A, B] = 9?

[spoiler]OA: {3, -3} this is from Manhattan guide 3, the answer alludes to factoring but doesn't show it...

Thanks!
[/spoiler]
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by goalevan » Sun Aug 07, 2011 2:11 pm
A^2 + 2AB + B^2 is the sum of squares formula, and equals (A + B)^2:

A^2 + 2AB + B^2
A^2 + AB + AB + B^2
A(A + B) + B(A + B)
(A + B)(A + B)
(A + B)^2

Therefore [A, B] = (A + B)^2

[A, B] = 9
(A + B)^2 = 9
A + B = -3 or 3
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by tpr-becky » Sun Aug 07, 2011 11:19 pm
This problem is better understood if you move the formula around to create A^2 + 2AB + B^2 - this is a common quadratic that is factored out to (A + B)(A + B).

If 9 = A^2 + B^2 + 2AB, then 9 = (A + B)(A + B) thus sqrt9 = A + B which means that A+B = 3 or -3
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