Evaluate

-\frac{y}{\left(y-1\right)\left(y+3\right)}

$−(y−1)(y+3)y $

Expand

-\frac{y}{\left(y-1\right)\left(y+3\right)}

$−(y−1)(y+3)y $

Graph

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\frac{2\left(y-3\right)}{\left(y-3\right)\left(y+3\right)}-\frac{y}{y-1}+\frac{y^{2}+2}{y^{2}+2y-3}

Factor the expressions that are not already factored in \frac{2y-6}{y^{2}-9}.

\frac{2}{y+3}-\frac{y}{y-1}+\frac{y^{2}+2}{y^{2}+2y-3}

Cancel out y-3 in both numerator and denominator.

\frac{2\left(y-1\right)}{\left(y-1\right)\left(y+3\right)}-\frac{y\left(y+3\right)}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y+3 and y-1 is \left(y-1\right)\left(y+3\right). Multiply \frac{2}{y+3} times \frac{y-1}{y-1}. Multiply \frac{y}{y-1} times \frac{y+3}{y+3}.

\frac{2\left(y-1\right)-y\left(y+3\right)}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}

Since \frac{2\left(y-1\right)}{\left(y-1\right)\left(y+3\right)} and \frac{y\left(y+3\right)}{\left(y-1\right)\left(y+3\right)} have the same denominator, subtract them by subtracting their numerators.

\frac{2y-2-y^{2}-3y}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}

Do the multiplications in 2\left(y-1\right)-y\left(y+3\right).

\frac{-y-2-y^{2}}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}

Combine like terms in 2y-2-y^{2}-3y.

\frac{-y-2-y^{2}}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{\left(y-1\right)\left(y+3\right)}

Factor y^{2}+2y-3.

\frac{-y-2-y^{2}+y^{2}+2}{\left(y-1\right)\left(y+3\right)}

Since \frac{-y-2-y^{2}}{\left(y-1\right)\left(y+3\right)} and \frac{y^{2}+2}{\left(y-1\right)\left(y+3\right)} have the same denominator, add them by adding their numerators.

\frac{-y}{\left(y-1\right)\left(y+3\right)}

Combine like terms in -y-2-y^{2}+y^{2}+2.

\frac{-y}{y^{2}+2y-3}

Expand \left(y-1\right)\left(y+3\right).

\frac{2\left(y-3\right)}{\left(y-3\right)\left(y+3\right)}-\frac{y}{y-1}+\frac{y^{2}+2}{y^{2}+2y-3}

Factor the expressions that are not already factored in \frac{2y-6}{y^{2}-9}.

\frac{2}{y+3}-\frac{y}{y-1}+\frac{y^{2}+2}{y^{2}+2y-3}

Cancel out y-3 in both numerator and denominator.

\frac{2\left(y-1\right)}{\left(y-1\right)\left(y+3\right)}-\frac{y\left(y+3\right)}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y+3 and y-1 is \left(y-1\right)\left(y+3\right). Multiply \frac{2}{y+3} times \frac{y-1}{y-1}. Multiply \frac{y}{y-1} times \frac{y+3}{y+3}.

\frac{2\left(y-1\right)-y\left(y+3\right)}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}

Since \frac{2\left(y-1\right)}{\left(y-1\right)\left(y+3\right)} and \frac{y\left(y+3\right)}{\left(y-1\right)\left(y+3\right)} have the same denominator, subtract them by subtracting their numerators.

\frac{2y-2-y^{2}-3y}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}

Do the multiplications in 2\left(y-1\right)-y\left(y+3\right).

\frac{-y-2-y^{2}}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{y^{2}+2y-3}

Combine like terms in 2y-2-y^{2}-3y.

\frac{-y-2-y^{2}}{\left(y-1\right)\left(y+3\right)}+\frac{y^{2}+2}{\left(y-1\right)\left(y+3\right)}

Factor y^{2}+2y-3.

\frac{-y-2-y^{2}+y^{2}+2}{\left(y-1\right)\left(y+3\right)}

Since \frac{-y-2-y^{2}}{\left(y-1\right)\left(y+3\right)} and \frac{y^{2}+2}{\left(y-1\right)\left(y+3\right)} have the same denominator, add them by adding their numerators.

\frac{-y}{\left(y-1\right)\left(y+3\right)}

Combine like terms in -y-2-y^{2}+y^{2}+2.

\frac{-y}{y^{2}+2y-3}

Expand \left(y-1\right)\left(y+3\right).

Examples

Quadratic equation

{ x } ^ { 2 } - 4 x - 5 = 0

$x_{2}−4x−5=0$

Trigonometry

4 \sin \theta \cos \theta = 2 \sin \theta

$4sinθcosθ=2sinθ$

Linear equation

y = 3x + 4

$y=3x+4$

Arithmetic

699 * 533

$699∗533$

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]

$[25 34 ][2−1 01 35 ]$

Simultaneous equation

\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.

${8x+2y=467x+3y=47 $

Differentiation

\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }

$dxd (x−5)(3x_{2}−2) $

Integration

\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x

$∫_{0}xe_{−x_{2}}dx$

Limits

\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}

$x→−3lim x_{2}+2x−3x_{2}−9 $