Evaluate
\frac{82+a-a^{2}}{a-2}
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\frac{82+a-a^{2}}{a-2}
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\left(\frac{3}{a+1}-\frac{a\left(a+1\right)}{a+1}+1\right)\times \frac{a+1}{\left(a-2\right)^{2}}+\frac{84}{a-2}-a
Divide a+1 by a+1 to get 1.
\left(\frac{3}{a+1}-a+1\right)\times \frac{a+1}{\left(a-2\right)^{2}}+\frac{84}{a-2}-a
Cancel out a+1 in both numerator and denominator.
\left(\frac{3}{a+1}+\frac{\left(-a+1\right)\left(a+1\right)}{a+1}\right)\times \frac{a+1}{\left(a-2\right)^{2}}+\frac{84}{a-2}-a
To add or subtract expressions, expand them to make their denominators the same. Multiply -a+1 times \frac{a+1}{a+1}.
\frac{3+\left(-a+1\right)\left(a+1\right)}{a+1}\times \frac{a+1}{\left(a-2\right)^{2}}+\frac{84}{a-2}-a
Since \frac{3}{a+1} and \frac{\left(-a+1\right)\left(a+1\right)}{a+1} have the same denominator, add them by adding their numerators.
\frac{3-a^{2}-a+a+1}{a+1}\times \frac{a+1}{\left(a-2\right)^{2}}+\frac{84}{a-2}-a
Do the multiplications in 3+\left(-a+1\right)\left(a+1\right).
\frac{4-a^{2}}{a+1}\times \frac{a+1}{\left(a-2\right)^{2}}+\frac{84}{a-2}-a
Combine like terms in 3-a^{2}-a+a+1.
\frac{\left(4-a^{2}\right)\left(a+1\right)}{\left(a+1\right)\left(a-2\right)^{2}}+\frac{84}{a-2}-a
Multiply \frac{4-a^{2}}{a+1} times \frac{a+1}{\left(a-2\right)^{2}} by multiplying numerator times numerator and denominator times denominator.
\frac{-a^{2}+4}{\left(a-2\right)^{2}}+\frac{84}{a-2}-a
Cancel out a+1 in both numerator and denominator.
\frac{-a^{2}+4}{\left(a-2\right)^{2}}+\frac{84\left(a-2\right)}{\left(a-2\right)^{2}}-a
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-2\right)^{2} and a-2 is \left(a-2\right)^{2}. Multiply \frac{84}{a-2} times \frac{a-2}{a-2}.
\frac{-a^{2}+4+84\left(a-2\right)}{\left(a-2\right)^{2}}-a
Since \frac{-a^{2}+4}{\left(a-2\right)^{2}} and \frac{84\left(a-2\right)}{\left(a-2\right)^{2}} have the same denominator, add them by adding their numerators.
\frac{-a^{2}+4+84a-168}{\left(a-2\right)^{2}}-a
Do the multiplications in -a^{2}+4+84\left(a-2\right).
\frac{-a^{2}-164+84a}{\left(a-2\right)^{2}}-a
Combine like terms in -a^{2}+4+84a-168.
\frac{-a^{2}-164+84a}{\left(a-2\right)^{2}}-\frac{a\left(a-2\right)^{2}}{\left(a-2\right)^{2}}
To add or subtract expressions, expand them to make their denominators the same. Multiply a times \frac{\left(a-2\right)^{2}}{\left(a-2\right)^{2}}.
\frac{-a^{2}-164+84a-a\left(a-2\right)^{2}}{\left(a-2\right)^{2}}
Since \frac{-a^{2}-164+84a}{\left(a-2\right)^{2}} and \frac{a\left(a-2\right)^{2}}{\left(a-2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.
\frac{-a^{2}-164+84a-a^{3}+4a^{2}-4a}{\left(a-2\right)^{2}}
Do the multiplications in -a^{2}-164+84a-a\left(a-2\right)^{2}.
\frac{3a^{2}-164+80a-a^{3}}{\left(a-2\right)^{2}}
Combine like terms in -a^{2}-164+84a-a^{3}+4a^{2}-4a.
\frac{\left(a-2\right)\left(-a^{2}+a+82\right)}{\left(a-2\right)^{2}}
Factor the expressions that are not already factored in \frac{3a^{2}-164+80a-a^{3}}{\left(a-2\right)^{2}}.
\frac{-a^{2}+a+82}{a-2}
Cancel out a-2 in both numerator and denominator.
\left(\frac{3}{a+1}-\frac{a\left(a+1\right)}{a+1}+1\right)\times \frac{a+1}{\left(a-2\right)^{2}}+\frac{84}{a-2}-a
Divide a+1 by a+1 to get 1.
\left(\frac{3}{a+1}-a+1\right)\times \frac{a+1}{\left(a-2\right)^{2}}+\frac{84}{a-2}-a
Cancel out a+1 in both numerator and denominator.
\left(\frac{3}{a+1}+\frac{\left(-a+1\right)\left(a+1\right)}{a+1}\right)\times \frac{a+1}{\left(a-2\right)^{2}}+\frac{84}{a-2}-a
To add or subtract expressions, expand them to make their denominators the same. Multiply -a+1 times \frac{a+1}{a+1}.
\frac{3+\left(-a+1\right)\left(a+1\right)}{a+1}\times \frac{a+1}{\left(a-2\right)^{2}}+\frac{84}{a-2}-a
Since \frac{3}{a+1} and \frac{\left(-a+1\right)\left(a+1\right)}{a+1} have the same denominator, add them by adding their numerators.
\frac{3-a^{2}-a+a+1}{a+1}\times \frac{a+1}{\left(a-2\right)^{2}}+\frac{84}{a-2}-a
Do the multiplications in 3+\left(-a+1\right)\left(a+1\right).
\frac{4-a^{2}}{a+1}\times \frac{a+1}{\left(a-2\right)^{2}}+\frac{84}{a-2}-a
Combine like terms in 3-a^{2}-a+a+1.
\frac{\left(4-a^{2}\right)\left(a+1\right)}{\left(a+1\right)\left(a-2\right)^{2}}+\frac{84}{a-2}-a
Multiply \frac{4-a^{2}}{a+1} times \frac{a+1}{\left(a-2\right)^{2}} by multiplying numerator times numerator and denominator times denominator.
\frac{-a^{2}+4}{\left(a-2\right)^{2}}+\frac{84}{a-2}-a
Cancel out a+1 in both numerator and denominator.
\frac{-a^{2}+4}{\left(a-2\right)^{2}}+\frac{84\left(a-2\right)}{\left(a-2\right)^{2}}-a
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-2\right)^{2} and a-2 is \left(a-2\right)^{2}. Multiply \frac{84}{a-2} times \frac{a-2}{a-2}.
\frac{-a^{2}+4+84\left(a-2\right)}{\left(a-2\right)^{2}}-a
Since \frac{-a^{2}+4}{\left(a-2\right)^{2}} and \frac{84\left(a-2\right)}{\left(a-2\right)^{2}} have the same denominator, add them by adding their numerators.
\frac{-a^{2}+4+84a-168}{\left(a-2\right)^{2}}-a
Do the multiplications in -a^{2}+4+84\left(a-2\right).
\frac{-a^{2}-164+84a}{\left(a-2\right)^{2}}-a
Combine like terms in -a^{2}+4+84a-168.
\frac{-a^{2}-164+84a}{\left(a-2\right)^{2}}-\frac{a\left(a-2\right)^{2}}{\left(a-2\right)^{2}}
To add or subtract expressions, expand them to make their denominators the same. Multiply a times \frac{\left(a-2\right)^{2}}{\left(a-2\right)^{2}}.
\frac{-a^{2}-164+84a-a\left(a-2\right)^{2}}{\left(a-2\right)^{2}}
Since \frac{-a^{2}-164+84a}{\left(a-2\right)^{2}} and \frac{a\left(a-2\right)^{2}}{\left(a-2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.
\frac{-a^{2}-164+84a-a^{3}+4a^{2}-4a}{\left(a-2\right)^{2}}
Do the multiplications in -a^{2}-164+84a-a\left(a-2\right)^{2}.
\frac{3a^{2}-164+80a-a^{3}}{\left(a-2\right)^{2}}
Combine like terms in -a^{2}-164+84a-a^{3}+4a^{2}-4a.
\frac{\left(a-2\right)\left(-a^{2}+a+82\right)}{\left(a-2\right)^{2}}
Factor the expressions that are not already factored in \frac{3a^{2}-164+80a-a^{3}}{\left(a-2\right)^{2}}.
\frac{-a^{2}+a+82}{a-2}
Cancel out a-2 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}