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## Algebraic Manipulation

Gauss, Ramanujan, and a pantheon of other mathematicians have given us algebraic manipulation tools way beyond what's taught in school. Learn what they knew.

# Rearranging

If $$\frac{1}{a}-\frac{1}{b}=-37$$, the value of $\frac{a+8ab-b}{a-3ab-b}$ can be expressed as $$\frac{n}{m}$$, where $$m$$ and $$n$$ are coprime positive integers. What is the value of $$m+n$$?

If the polynomial $x^2-y^2+11x+7y+18$ can be factorized as $$(x+y+a)(x-y+b),$$ what is the value of $$a+b?$$

If $$-x^3+6x^2-6x+36$$ is factorized as $$(a-x)(x^2+b),$$ what is the value of $$a+b$$?

If $$x+2y-6+3(1-x-2y)=-51$$, what is $$x+2y$$?

If $$x+2y=-13$$, what is the value of $x^2+4xy-2x-4y-3+4y^2?$

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