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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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