Evaluate
-\frac{1}{2-x}
Expand
\frac{1}{x-2}
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\frac{x}{\left(x-1\right)\left(x+2\right)}-\frac{3}{\left(x-2\right)\left(-x-2\right)}+\frac{1}{\left(x-1\right)\left(x+2\right)\left(x-2\right)}
Factor 4-x^{2}.
\frac{x\left(x-2\right)}{\left(x-2\right)\left(x-1\right)\left(x+2\right)}-\frac{3\left(-1\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)\left(x+2\right)}+\frac{1}{\left(x-1\right)\left(x+2\right)\left(x-2\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+2\right) and \left(x-2\right)\left(-x-2\right) is \left(x-2\right)\left(x-1\right)\left(x+2\right). Multiply \frac{x}{\left(x-1\right)\left(x+2\right)} times \frac{x-2}{x-2}. Multiply \frac{3}{\left(x-2\right)\left(-x-2\right)} times \frac{-\left(x-1\right)}{-\left(x-1\right)}.
\frac{x\left(x-2\right)-3\left(-1\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)\left(x+2\right)}+\frac{1}{\left(x-1\right)\left(x+2\right)\left(x-2\right)}
Since \frac{x\left(x-2\right)}{\left(x-2\right)\left(x-1\right)\left(x+2\right)} and \frac{3\left(-1\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{x^{2}-2x+3x-3}{\left(x-2\right)\left(x-1\right)\left(x+2\right)}+\frac{1}{\left(x-1\right)\left(x+2\right)\left(x-2\right)}
Do the multiplications in x\left(x-2\right)-3\left(-1\right)\left(x-1\right).
\frac{x^{2}+x-3}{\left(x-2\right)\left(x-1\right)\left(x+2\right)}+\frac{1}{\left(x-1\right)\left(x+2\right)\left(x-2\right)}
Combine like terms in x^{2}-2x+3x-3.
\frac{x^{2}+x-3+1}{\left(x-2\right)\left(x-1\right)\left(x+2\right)}
Since \frac{x^{2}+x-3}{\left(x-2\right)\left(x-1\right)\left(x+2\right)} and \frac{1}{\left(x-1\right)\left(x+2\right)\left(x-2\right)} have the same denominator, add them by adding their numerators.
\frac{x^{2}+x-2}{\left(x-2\right)\left(x-1\right)\left(x+2\right)}
Combine like terms in x^{2}+x-3+1.
\frac{\left(x-1\right)\left(x+2\right)}{\left(x-2\right)\left(x-1\right)\left(x+2\right)}
Factor the expressions that are not already factored in \frac{x^{2}+x-2}{\left(x-2\right)\left(x-1\right)\left(x+2\right)}.
\frac{1}{x-2}
Cancel out \left(x-1\right)\left(x+2\right) in both numerator and denominator.
\frac{x}{\left(x-1\right)\left(x+2\right)}-\frac{3}{\left(x-2\right)\left(-x-2\right)}+\frac{1}{\left(x-1\right)\left(x+2\right)\left(x-2\right)}
Factor 4-x^{2}.
\frac{x\left(x-2\right)}{\left(x-2\right)\left(x-1\right)\left(x+2\right)}-\frac{3\left(-1\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)\left(x+2\right)}+\frac{1}{\left(x-1\right)\left(x+2\right)\left(x-2\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+2\right) and \left(x-2\right)\left(-x-2\right) is \left(x-2\right)\left(x-1\right)\left(x+2\right). Multiply \frac{x}{\left(x-1\right)\left(x+2\right)} times \frac{x-2}{x-2}. Multiply \frac{3}{\left(x-2\right)\left(-x-2\right)} times \frac{-\left(x-1\right)}{-\left(x-1\right)}.
\frac{x\left(x-2\right)-3\left(-1\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)\left(x+2\right)}+\frac{1}{\left(x-1\right)\left(x+2\right)\left(x-2\right)}
Since \frac{x\left(x-2\right)}{\left(x-2\right)\left(x-1\right)\left(x+2\right)} and \frac{3\left(-1\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{x^{2}-2x+3x-3}{\left(x-2\right)\left(x-1\right)\left(x+2\right)}+\frac{1}{\left(x-1\right)\left(x+2\right)\left(x-2\right)}
Do the multiplications in x\left(x-2\right)-3\left(-1\right)\left(x-1\right).
\frac{x^{2}+x-3}{\left(x-2\right)\left(x-1\right)\left(x+2\right)}+\frac{1}{\left(x-1\right)\left(x+2\right)\left(x-2\right)}
Combine like terms in x^{2}-2x+3x-3.
\frac{x^{2}+x-3+1}{\left(x-2\right)\left(x-1\right)\left(x+2\right)}
Since \frac{x^{2}+x-3}{\left(x-2\right)\left(x-1\right)\left(x+2\right)} and \frac{1}{\left(x-1\right)\left(x+2\right)\left(x-2\right)} have the same denominator, add them by adding their numerators.
\frac{x^{2}+x-2}{\left(x-2\right)\left(x-1\right)\left(x+2\right)}
Combine like terms in x^{2}+x-3+1.
\frac{\left(x-1\right)\left(x+2\right)}{\left(x-2\right)\left(x-1\right)\left(x+2\right)}
Factor the expressions that are not already factored in \frac{x^{2}+x-2}{\left(x-2\right)\left(x-1\right)\left(x+2\right)}.
\frac{1}{x-2}
Cancel out \left(x-1\right)\left(x+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}