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I'm trying to find out how to solve this:

$\frac{x-2}{x^2+2x} - \frac{x+2}{x^2-2x} - \frac{4x}{x^2-4}$

The answer is $\displaystyle \frac{-4}{x-2}$

What is this called? And is there any video of it on http://www.khanacademy.org ? Thanks!

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    @J.M. I commented in the meta thread, giving two examples of (apparently) different criteria, this one and [Simplification: $\biggl(\frac{ 1+x^2}{1-x^2}\biggr)^2 = \frac{1}{1-y^2}$](http://math.stackexchange.com/q/11217/752).2011-09-15

1 Answers 1

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Added: The method is to transform the sum of the given rational fractions (the numerator and denominator consists of polynomials) into a single equivalent fraction.

The properties used are

  1. $\frac{A(x)}{B(x)}=\frac{A(x)P(x)}{B(x)P(x)}\qquad\text{for }P(x)\neq 0.$

  2. $\frac{A(x)}{B(x)}\pm \frac{C(x)}{D(x)}=\frac{A(x)D(x)\pm B(x)C(x)}{B(x)D(x)}.$


I would calculate as follows, starting with the factorization of denominators

$\begin{eqnarray*} &&\frac{x-2}{x^{2}+2x}-\frac{x+2}{x^{2}-2x}-\frac{4x}{x^{2}-4} \\ &=&\frac{x-2}{x(x+2)}-\frac{x+2}{x\left( x-2\right) }-\frac{4x}{\left( x-2\right) \left( x+2\right) } \\ &=&\frac{\left( x-2\right) ^{2}}{x(x+2)\left( x-2\right) }-\frac{\left( x+2\right) ^{2}}{x\left( x-2\right) \left( x+2\right) }-\frac{4x^{2}}{% x\left( x-2\right) \left( x+2\right) } \\ &=&\frac{\left( x-2\right) ^{2}-\left( x+2\right) ^{2}-4x^{2}}{x\left( x-2\right) \left( x+2\right) } \\ &=&\frac{x^{2}-4x+4-x^{2}-4x-4-4x^{2}}{x\left( x-2\right) \left( x+2\right) } \\ &=&\frac{-4x^{2}-8x}{x\left( x-2\right) \left( x+2\right) } \\ &=&-\frac{4x\left( x+2\right) }{x\left( x-2\right) \left( x+2\right) } \\ &=&-\frac{4}{x-2}. \end{eqnarray*}$

so that I reduce the rational fractions to a common denominator first to allow me to add (subtract) them, and simplify the numerator thereafter. Finally I divide by the common factors to both the numerator and denominator. This is valid iff $x\ne -2$ and $x\ne 0$, because you cannot divide by zero. Also for $x=2$ the fraction is not defined.

See "Simplifying Rational Expressions 1, 2 and 3", "Adding and Subtracting Rational Expressions 1, 2 and 3" on http://www.khanacademy.org.

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    @Muazam: You are welcome!2011-05-28