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Question
Simplify
((a^2)/((a-b)^2)) - (b/(a-b))-(a/(b-a))
Okay, put everything over a common denominator.
Observe the following:
(a-b)^2=(a-b)(a-b)
Let fraction F1=((a^2)/((a-b)^2)),
F2=(b/(a-b)),F3=(a/(b-a)).
The given expression ((a^2)/((a-b)^2)) - (b/(a-b))-(a/(b-a)) is same as F1-F2-F3.
F1-F2-F3
=F1-F2*(a-b)/(a-b)-F3*(a-b)/(a-b) ...agree?
=F1-F2*(a-b)/(a-b)+(-F3*(a-b))/(a-b)
...since (b-a)=-(a-b) in the denominator of F3
=[a^2-b(a-b)+a(a-b)]/[(a-b)^2]
=[a^2-ab+b^2+a^2-ab]/[(a-b)^2]
=[a^2-2ab+b^2 +a^2]/[(a-b)^2]
=[(a-b)^2+a^2]/[(a-b)^2]
=1 + a^2/[(a-b)^2]
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Josh
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When I work through problems, I like to emphasize concepts which I believe are worth noting. I will try to answer questions in the following areas, but not at the advanced level. Algebra. Sequences & Series. Trigonometry. Functions & Graphs. Coordinate Geometry. Quadratic Polynomials. Exponential & Logarithms. Basic Calculus. Probability, Permutation and Combination. Mathematical Induction. Complex numbers. Physics problems.
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