# How do you simplify (a+2b)/(x+a)-(a-2b)/(x-a)-(4bx-2a^2)/(x^2-a^2)?

Aug 15, 2017

$= 0$

#### Explanation:

$\frac{a + 2 b}{x + a} - \frac{a - 2 b}{x - a} - \frac{4 b x - 2 {a}^{2}}{{x}^{2} - {a}^{2}} \text{ } \leftarrow$ factorise

$\frac{a + 2 b}{x + a} - \frac{a - 2 b}{x - a} - \frac{\left(4 b x - 2 {a}^{2}\right)}{\left(x + a\right) \left(x - a\right)}$

Find a common denominator and equivalent fractions:

$\frac{a + 2 b}{x + a} \times \textcolor{b l u e}{\frac{x - a}{x - a}} - \frac{a - 2 b}{x - a} \times \textcolor{b l u e}{\frac{x + a}{x + a}} - \frac{\left(4 b x - 2 {a}^{2}\right)}{\left(x + a\right) \left(x - a\right)}$

$\frac{\left(a + 2 b\right) \left(x - a\right) - \left(a - 2 b\right) \left(x + a\right) - \left(4 b x - 2 {a}^{2}\right)}{\left(x + a\right) \left(x - a\right)}$

$= \frac{\left(a x - {a}^{2} + 2 b x - 2 a b\right) - \left(a x + {a}^{2} - 2 b x - 2 a b\right) - \left(4 b x - 2 {a}^{2}\right)}{\left(x + a\right) \left(x - a\right)}$

$= \frac{\textcolor{b l u e}{a x} \textcolor{red}{- {a}^{2}} \textcolor{f \mathmr{and} e s t g r e e n}{+ 2 b x} \textcolor{m a \ge n t a}{- 2 a b} \textcolor{b l u e}{- a x} \textcolor{red}{- {a}^{2}} \textcolor{f \mathmr{and} e s t g r e e n}{+ 2 b x} \textcolor{m a \ge n t a}{+ 2 a b} \textcolor{f \mathmr{and} e s t g r e e n}{- 4 b x} \textcolor{red}{+ 2 {a}^{2}}}{\left(x + a\right) \left(x - a\right)}$

$= \frac{\textcolor{b l u e}{0 a x} \textcolor{red}{+ 0 {a}^{2}} \textcolor{f \mathmr{and} e s t g r e e n}{+ 0 b x} \textcolor{m a \ge n t a}{+ 0 a b}}{\left(x + a\right) \left(x - a\right)}$

$= 0$

Aug 15, 2017

$0$

#### Explanation:

$\text{before we can add/subtract fractions we require them to}$
$\text{have a "color(blue)"common denominator}$

${x}^{2} - {a}^{2} \text{ is a "color(blue)"difference of squares}$

$\Rightarrow {a}^{2} - {x}^{2} = \left(x - a\right) \left(x + a\right)$

$\text{multiply numerator/denominator of "(a+2b)/(x+a)" by } \left(x - a\right)$

$\text{multiply numerator/denominator of "(a-2b)/(x-a)" by } \left(x + a\right)$

$\frac{\left(a + 2 b\right) \left(x - a\right)}{\left(x - a\right) \left(x + a\right)} - \frac{\left(a - 2 b\right) \left(x + a\right)}{\left(x - a\right) \left(x + a\right)} - \frac{4 b x - 2 {a}^{2}}{\left(x - a\right) \left(x + a\right)}$

$\text{now add/subtract the numerators leaving the denominator}$
$\text{first multiplying out the brackets}$

$\frac{a x + 2 b x - {a}^{2} - 2 a b - \left(a x - 2 b x + {a}^{2} - 2 a b\right) - \left(4 b x - 2 {a}^{2}\right)}{\left(x - a\right) \left(x + a\right)}$

$= \frac{a x + 2 b x - {a}^{2} - 2 a b - a x + 2 b x - {a}^{2} + 2 a b - 4 b x + 2 {a}^{2}}{\left(x - a\right) \left(x + a\right)}$

$= 0$