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\frac{1}{a+2}
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\frac{1}{a+2}
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\frac{\frac{a+2}{a\left(a-2\right)}+\frac{8}{\left(a-2\right)\left(-a-2\right)}}{\frac{a-2}{a}}
Factor a^{2}-2a. Factor 4-a^{2}.
\frac{\frac{\left(a+2\right)\left(-a-2\right)}{a\left(a-2\right)\left(-a-2\right)}+\frac{8a}{a\left(a-2\right)\left(-a-2\right)}}{\frac{a-2}{a}}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\left(a-2\right) and \left(a-2\right)\left(-a-2\right) is a\left(a-2\right)\left(-a-2\right). Multiply \frac{a+2}{a\left(a-2\right)} times \frac{-a-2}{-a-2}. Multiply \frac{8}{\left(a-2\right)\left(-a-2\right)} times \frac{a}{a}.
\frac{\frac{\left(a+2\right)\left(-a-2\right)+8a}{a\left(a-2\right)\left(-a-2\right)}}{\frac{a-2}{a}}
Since \frac{\left(a+2\right)\left(-a-2\right)}{a\left(a-2\right)\left(-a-2\right)} and \frac{8a}{a\left(a-2\right)\left(-a-2\right)} have the same denominator, add them by adding their numerators.
\frac{\frac{-a^{2}-2a-2a-4+8a}{a\left(a-2\right)\left(-a-2\right)}}{\frac{a-2}{a}}
Do the multiplications in \left(a+2\right)\left(-a-2\right)+8a.
\frac{\frac{-a^{2}+4a-4}{a\left(a-2\right)\left(-a-2\right)}}{\frac{a-2}{a}}
Combine like terms in -a^{2}-2a-2a-4+8a.
\frac{\frac{\left(a-2\right)\left(-a+2\right)}{a\left(a-2\right)\left(-a-2\right)}}{\frac{a-2}{a}}
Factor the expressions that are not already factored in \frac{-a^{2}+4a-4}{a\left(a-2\right)\left(-a-2\right)}.
\frac{\frac{-\left(a-2\right)\left(a-2\right)}{a\left(a-2\right)\left(-a-2\right)}}{\frac{a-2}{a}}
Extract the negative sign in 2-a.
\frac{\frac{-\left(a-2\right)}{a\left(-a-2\right)}}{\frac{a-2}{a}}
Cancel out a-2 in both numerator and denominator.
\frac{-\left(a-2\right)a}{a\left(-a-2\right)\left(a-2\right)}
Divide \frac{-\left(a-2\right)}{a\left(-a-2\right)} by \frac{a-2}{a} by multiplying \frac{-\left(a-2\right)}{a\left(-a-2\right)} by the reciprocal of \frac{a-2}{a}.
\frac{-1}{-a-2}
Cancel out a\left(a-2\right) in both numerator and denominator.
\frac{\frac{a+2}{a\left(a-2\right)}+\frac{8}{\left(a-2\right)\left(-a-2\right)}}{\frac{a-2}{a}}
Factor a^{2}-2a. Factor 4-a^{2}.
\frac{\frac{\left(a+2\right)\left(-a-2\right)}{a\left(a-2\right)\left(-a-2\right)}+\frac{8a}{a\left(a-2\right)\left(-a-2\right)}}{\frac{a-2}{a}}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\left(a-2\right) and \left(a-2\right)\left(-a-2\right) is a\left(a-2\right)\left(-a-2\right). Multiply \frac{a+2}{a\left(a-2\right)} times \frac{-a-2}{-a-2}. Multiply \frac{8}{\left(a-2\right)\left(-a-2\right)} times \frac{a}{a}.
\frac{\frac{\left(a+2\right)\left(-a-2\right)+8a}{a\left(a-2\right)\left(-a-2\right)}}{\frac{a-2}{a}}
Since \frac{\left(a+2\right)\left(-a-2\right)}{a\left(a-2\right)\left(-a-2\right)} and \frac{8a}{a\left(a-2\right)\left(-a-2\right)} have the same denominator, add them by adding their numerators.
\frac{\frac{-a^{2}-2a-2a-4+8a}{a\left(a-2\right)\left(-a-2\right)}}{\frac{a-2}{a}}
Do the multiplications in \left(a+2\right)\left(-a-2\right)+8a.
\frac{\frac{-a^{2}+4a-4}{a\left(a-2\right)\left(-a-2\right)}}{\frac{a-2}{a}}
Combine like terms in -a^{2}-2a-2a-4+8a.
\frac{\frac{\left(a-2\right)\left(-a+2\right)}{a\left(a-2\right)\left(-a-2\right)}}{\frac{a-2}{a}}
Factor the expressions that are not already factored in \frac{-a^{2}+4a-4}{a\left(a-2\right)\left(-a-2\right)}.
\frac{\frac{-\left(a-2\right)\left(a-2\right)}{a\left(a-2\right)\left(-a-2\right)}}{\frac{a-2}{a}}
Extract the negative sign in 2-a.
\frac{\frac{-\left(a-2\right)}{a\left(-a-2\right)}}{\frac{a-2}{a}}
Cancel out a-2 in both numerator and denominator.
\frac{-\left(a-2\right)a}{a\left(-a-2\right)\left(a-2\right)}
Divide \frac{-\left(a-2\right)}{a\left(-a-2\right)} by \frac{a-2}{a} by multiplying \frac{-\left(a-2\right)}{a\left(-a-2\right)} by the reciprocal of \frac{a-2}{a}.
\frac{-1}{-a-2}
Cancel out a\left(a-2\right) in both numerator and denominator.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}