[tex] \huge \boxed{\mathbb{QUESTION} \downarrow}[/tex]
- Simplify ⇨ 1/x(x+a) + 1/x(x-a)
[tex] \large \boxed{\mathbb{ANSWER \: WITH \: EXPLANATION} \downarrow}[/tex]
[tex] \sf\frac { 1 } { x ( x + a ) } + \frac { 1 } { x ( x - a ) } \\ [/tex]
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of [tex]x\left(x+a\right)[/tex] and [tex]x\left(x-a\right)[/tex] is [tex]x\left(x+a\right)\left(x-a\right)[/tex]. Multiply [tex]\frac{1}{x\left(x+a\right)} times \frac{x-a}{x-a}[/tex]. Multiply [tex]\frac{1}{x\left(x-a\right)} times \frac{x+a}{x+a}[/tex].
[tex] \sf\frac{x-a}{x\left(x+a\right)\left(x-a\right)}+\frac{x+a}{x\left(x+a\right)\left(x-a\right)} \\ [/tex]
Because [tex]\frac{x-a}{x\left(x+a\right)\left(x-a\right)}[/tex] and [tex]\frac{x+a}{x\left(x+a\right)\left(x-a\right)}[/tex] have the same denominator, add them by adding their numerators.
[tex] \sf\frac{x-a+x+a}{x\left(x+a\right)\left(x-a\right)} \\ [/tex]
Combine like terms in x-a+x+a.
[tex] \sf\frac{2x}{x\left(x+a\right)\left(x-a\right)} \\ [/tex]
Cancel out x in both the numerator and denominator.
[tex] \sf\frac{2}{\left(x+a\right)\left(x-a\right)} \\ [/tex]
Expand [tex]\left(x+a\right)\left(x-a\right)[/tex].
[tex] \boxed{\boxed{ \bf\frac{2}{x^{2}-a^{2}}}} \\ [/tex]
