Evaluate
-\frac{\left(a-2\right)\left(a^{2}+a+2\right)}{2a\left(a^{2}-1\right)}
Expand
-\frac{a^{3}-a^{2}-4}{2a\left(a^{2}-1\right)}
Share
Copied to clipboard
\frac{a-1}{a\left(a+1\right)}-\frac{a-3}{\left(a-1\right)\left(a+1\right)}-\frac{a^{2}+2}{2a^{2}+2a}
Factor a^{2}+a. Factor a^{2}-1.
\frac{\left(a-1\right)\left(a-1\right)}{a\left(a-1\right)\left(a+1\right)}-\frac{\left(a-3\right)a}{a\left(a-1\right)\left(a+1\right)}-\frac{a^{2}+2}{2a^{2}+2a}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\left(a+1\right) and \left(a-1\right)\left(a+1\right) is a\left(a-1\right)\left(a+1\right). Multiply \frac{a-1}{a\left(a+1\right)} times \frac{a-1}{a-1}. Multiply \frac{a-3}{\left(a-1\right)\left(a+1\right)} times \frac{a}{a}.
\frac{\left(a-1\right)\left(a-1\right)-\left(a-3\right)a}{a\left(a-1\right)\left(a+1\right)}-\frac{a^{2}+2}{2a^{2}+2a}
Since \frac{\left(a-1\right)\left(a-1\right)}{a\left(a-1\right)\left(a+1\right)} and \frac{\left(a-3\right)a}{a\left(a-1\right)\left(a+1\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{a^{2}-a-a+1-a^{2}+3a}{a\left(a-1\right)\left(a+1\right)}-\frac{a^{2}+2}{2a^{2}+2a}
Do the multiplications in \left(a-1\right)\left(a-1\right)-\left(a-3\right)a.
\frac{a+1}{a\left(a-1\right)\left(a+1\right)}-\frac{a^{2}+2}{2a^{2}+2a}
Combine like terms in a^{2}-a-a+1-a^{2}+3a.
\frac{1}{a\left(a-1\right)}-\frac{a^{2}+2}{2a^{2}+2a}
Cancel out a+1 in both numerator and denominator.
\frac{1}{a\left(a-1\right)}-\frac{a^{2}+2}{2a\left(a+1\right)}
Factor 2a^{2}+2a.
\frac{2\left(a+1\right)}{2a\left(a-1\right)\left(a+1\right)}-\frac{\left(a^{2}+2\right)\left(a-1\right)}{2a\left(a-1\right)\left(a+1\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\left(a-1\right) and 2a\left(a+1\right) is 2a\left(a-1\right)\left(a+1\right). Multiply \frac{1}{a\left(a-1\right)} times \frac{2\left(a+1\right)}{2\left(a+1\right)}. Multiply \frac{a^{2}+2}{2a\left(a+1\right)} times \frac{a-1}{a-1}.
\frac{2\left(a+1\right)-\left(a^{2}+2\right)\left(a-1\right)}{2a\left(a-1\right)\left(a+1\right)}
Since \frac{2\left(a+1\right)}{2a\left(a-1\right)\left(a+1\right)} and \frac{\left(a^{2}+2\right)\left(a-1\right)}{2a\left(a-1\right)\left(a+1\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{2a+2-a^{3}+a^{2}-2a+2}{2a\left(a-1\right)\left(a+1\right)}
Do the multiplications in 2\left(a+1\right)-\left(a^{2}+2\right)\left(a-1\right).
\frac{4-a^{3}+a^{2}}{2a\left(a-1\right)\left(a+1\right)}
Combine like terms in 2a+2-a^{3}+a^{2}-2a+2.
\frac{4-a^{3}+a^{2}}{2a^{3}-2a}
Expand 2a\left(a-1\right)\left(a+1\right).
\frac{a-1}{a\left(a+1\right)}-\frac{a-3}{\left(a-1\right)\left(a+1\right)}-\frac{a^{2}+2}{2a^{2}+2a}
Factor a^{2}+a. Factor a^{2}-1.
\frac{\left(a-1\right)\left(a-1\right)}{a\left(a-1\right)\left(a+1\right)}-\frac{\left(a-3\right)a}{a\left(a-1\right)\left(a+1\right)}-\frac{a^{2}+2}{2a^{2}+2a}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\left(a+1\right) and \left(a-1\right)\left(a+1\right) is a\left(a-1\right)\left(a+1\right). Multiply \frac{a-1}{a\left(a+1\right)} times \frac{a-1}{a-1}. Multiply \frac{a-3}{\left(a-1\right)\left(a+1\right)} times \frac{a}{a}.
\frac{\left(a-1\right)\left(a-1\right)-\left(a-3\right)a}{a\left(a-1\right)\left(a+1\right)}-\frac{a^{2}+2}{2a^{2}+2a}
Since \frac{\left(a-1\right)\left(a-1\right)}{a\left(a-1\right)\left(a+1\right)} and \frac{\left(a-3\right)a}{a\left(a-1\right)\left(a+1\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{a^{2}-a-a+1-a^{2}+3a}{a\left(a-1\right)\left(a+1\right)}-\frac{a^{2}+2}{2a^{2}+2a}
Do the multiplications in \left(a-1\right)\left(a-1\right)-\left(a-3\right)a.
\frac{a+1}{a\left(a-1\right)\left(a+1\right)}-\frac{a^{2}+2}{2a^{2}+2a}
Combine like terms in a^{2}-a-a+1-a^{2}+3a.
\frac{1}{a\left(a-1\right)}-\frac{a^{2}+2}{2a^{2}+2a}
Cancel out a+1 in both numerator and denominator.
\frac{1}{a\left(a-1\right)}-\frac{a^{2}+2}{2a\left(a+1\right)}
Factor 2a^{2}+2a.
\frac{2\left(a+1\right)}{2a\left(a-1\right)\left(a+1\right)}-\frac{\left(a^{2}+2\right)\left(a-1\right)}{2a\left(a-1\right)\left(a+1\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\left(a-1\right) and 2a\left(a+1\right) is 2a\left(a-1\right)\left(a+1\right). Multiply \frac{1}{a\left(a-1\right)} times \frac{2\left(a+1\right)}{2\left(a+1\right)}. Multiply \frac{a^{2}+2}{2a\left(a+1\right)} times \frac{a-1}{a-1}.
\frac{2\left(a+1\right)-\left(a^{2}+2\right)\left(a-1\right)}{2a\left(a-1\right)\left(a+1\right)}
Since \frac{2\left(a+1\right)}{2a\left(a-1\right)\left(a+1\right)} and \frac{\left(a^{2}+2\right)\left(a-1\right)}{2a\left(a-1\right)\left(a+1\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{2a+2-a^{3}+a^{2}-2a+2}{2a\left(a-1\right)\left(a+1\right)}
Do the multiplications in 2\left(a+1\right)-\left(a^{2}+2\right)\left(a-1\right).
\frac{4-a^{3}+a^{2}}{2a\left(a-1\right)\left(a+1\right)}
Combine like terms in 2a+2-a^{3}+a^{2}-2a+2.
\frac{4-a^{3}+a^{2}}{2a^{3}-2a}
Expand 2a\left(a-1\right)\left(a+1\right).
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}