Basic Math Examples

$\frac{3}{y^{2} + 3 y} - \frac{1}{y} - \frac{6}{y^{2} - 9}$

Step 1

Simplify each term.

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Step 1.1

Factor $y$ out of $y^{2} + 3 y$ .

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Step 1.1.1

Factor $y$ out of $y^{2}$ .

$\frac{3}{y \cdot y + 3 y} - \frac{1}{y} - \frac{6}{y^{2} - 9}$

Step 1.1.2

Factor $y$ out of $3 y$ .

$\frac{3}{y \cdot y + y \cdot 3} - \frac{1}{y} - \frac{6}{y^{2} - 9}$

Step 1.1.3

Factor $y$ out of $y \cdot y + y \cdot 3$ .

$\frac{3}{y (y + 3)} - \frac{1}{y} - \frac{6}{y^{2} - 9}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Rewrite $9$ as $3^{2}$ .

$\frac{3}{y (y + 3)} - \frac{1}{y} - \frac{6}{y^{2} - 3^{2}}$

Step 1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 3$ .

$\frac{3}{y (y + 3)} - \frac{1}{y} - \frac{6}{(y + 3) (y - 3)}$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{3}{y (y + 3)}$ by $\frac{y - 3}{y - 3}$ .

$\frac{3}{y (y + 3)} \cdot \frac{y - 3}{y - 3} - \frac{1}{y} - \frac{6}{(y + 3) (y - 3)}$

Step 2.2

Multiply $\frac{3}{y (y + 3)}$ by $\frac{y - 3}{y - 3}$ .

$\frac{3 (y - 3)}{y (y + 3) (y - 3)} - \frac{1}{y} - \frac{6}{(y + 3) (y - 3)}$

Step 2.3

Multiply $\frac{1}{y}$ by $\frac{(y + 3) (y - 3)}{(y + 3) (y - 3)}$ .

$\frac{3 (y - 3)}{y (y + 3) (y - 3)} - (\frac{1}{y} \cdot \frac{(y + 3) (y - 3)}{(y + 3) (y - 3)}) - \frac{6}{(y + 3) (y - 3)}$

Step 2.4

Multiply $\frac{1}{y}$ by $\frac{(y + 3) (y - 3)}{(y + 3) (y - 3)}$ .

$\frac{3 (y - 3)}{y (y + 3) (y - 3)} - \frac{(y + 3) (y - 3)}{y ((y + 3) (y - 3))} - \frac{6}{(y + 3) (y - 3)}$

Step 2.5

Multiply $\frac{6}{(y + 3) (y - 3)}$ by $\frac{y}{y}$ .

$\frac{3 (y - 3)}{y (y + 3) (y - 3)} - \frac{(y + 3) (y - 3)}{y ((y + 3) (y - 3))} - (\frac{6}{(y + 3) (y - 3)} \cdot \frac{y}{y})$

Step 2.6

Multiply $\frac{6}{(y + 3) (y - 3)}$ by $\frac{y}{y}$ .

$\frac{3 (y - 3)}{y (y + 3) (y - 3)} - \frac{(y + 3) (y - 3)}{y ((y + 3) (y - 3))} - \frac{6 y}{(y + 3) (y - 3) y}$

Step 2.7

Reorder the factors of $y ((y + 3) (y - 3))$ .

$\frac{3 (y - 3)}{y (y + 3) (y - 3)} - \frac{(y + 3) (y - 3)}{y (y + 3) (y - 3)} - \frac{6 y}{(y + 3) (y - 3) y}$

Step 2.8

Reorder the factors of $(y + 3) (y - 3) y$ .

$\frac{3 (y - 3)}{y (y + 3) (y - 3)} - \frac{(y + 3) (y - 3)}{y (y + 3) (y - 3)} - \frac{6 y}{y (y + 3) (y - 3)}$

Step 3

Combine the numerators over the common denominator.

$\frac{3 (y - 3) - (y + 3) (y - 3) - 6 y}{y (y + 3) (y - 3)}$

Step 4

Simplify each term.

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Step 4.1

Apply the distributive property.

$\frac{3 y + 3 \cdot - 3 - (y + 3) (y - 3) - 6 y}{y (y + 3) (y - 3)}$

Step 4.2

Multiply $3$ by $- 3$ .

$\frac{3 y - 9 - (y + 3) (y - 3) - 6 y}{y (y + 3) (y - 3)}$

Step 4.3

Apply the distributive property.

$\frac{3 y - 9 + (- y - 1 \cdot 3) (y - 3) - 6 y}{y (y + 3) (y - 3)}$

Step 4.4

Multiply $- 1$ by $3$ .

$\frac{3 y - 9 + (- y - 3) (y - 3) - 6 y}{y (y + 3) (y - 3)}$

Step 4.5

Expand $(- y - 3) (y - 3)$ using the FOIL Method.

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Step 4.5.1

Apply the distributive property.

$\frac{3 y - 9 - y (y - 3) - 3 (y - 3) - 6 y}{y (y + 3) (y - 3)}$

Step 4.5.2

Apply the distributive property.

$\frac{3 y - 9 - y \cdot y - y \cdot - 3 - 3 (y - 3) - 6 y}{y (y + 3) (y - 3)}$

Step 4.5.3

Apply the distributive property.

$\frac{3 y - 9 - y \cdot y - y \cdot - 3 - 3 y - 3 \cdot - 3 - 6 y}{y (y + 3) (y - 3)}$

Step 4.6

Simplify and combine like terms.

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Step 4.6.1

Simplify each term.

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Step 4.6.1.1

Multiply $y$ by $y$ by adding the exponents.

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Step 4.6.1.1.1

Move $y$ .

$\frac{3 y - 9 - (y \cdot y) - y \cdot - 3 - 3 y - 3 \cdot - 3 - 6 y}{y (y + 3) (y - 3)}$

Step 4.6.1.1.2

Multiply $y$ by $y$ .

$\frac{3 y - 9 - y^{2} - y \cdot - 3 - 3 y - 3 \cdot - 3 - 6 y}{y (y + 3) (y - 3)}$

Step 4.6.1.2

Multiply $- 3$ by $- 1$ .

$\frac{3 y - 9 - y^{2} + 3 y - 3 y - 3 \cdot - 3 - 6 y}{y (y + 3) (y - 3)}$

Step 4.6.1.3

Multiply $- 3$ by $- 3$ .

$\frac{3 y - 9 - y^{2} + 3 y - 3 y + 9 - 6 y}{y (y + 3) (y - 3)}$

Step 4.6.2

Subtract $3 y$ from $3 y$ .

$\frac{3 y - 9 - y^{2} + 0 + 9 - 6 y}{y (y + 3) (y - 3)}$

Step 4.6.3

Add $- y^{2}$ and $0$ .

$\frac{3 y - 9 - y^{2} + 9 - 6 y}{y (y + 3) (y - 3)}$

Step 5

Simplify terms.

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Step 5.1

Combine the opposite terms in $3 y - 9 - y^{2} + 9 - 6 y$ .

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Step 5.1.1

Add $- 9$ and $9$ .

$\frac{3 y - y^{2} + 0 - 6 y}{y (y + 3) (y - 3)}$

Step 5.1.2

Add $3 y - y^{2}$ and $0$ .

$\frac{3 y - y^{2} - 6 y}{y (y + 3) (y - 3)}$

Step 5.2

Subtract $6 y$ from $3 y$ .

$\frac{- y^{2} - 3 y}{y (y + 3) (y - 3)}$

Step 5.3

Factor $y$ out of $- y^{2} - 3 y$ .

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Step 5.3.1

Factor $y$ out of $- y^{2}$ .

$\frac{y (- y) - 3 y}{y (y + 3) (y - 3)}$

Step 5.3.2

Factor $y$ out of $- 3 y$ .

$\frac{y (- y) + y \cdot - 3}{y (y + 3) (y - 3)}$

Step 5.3.3

Factor $y$ out of $y (- y) + y \cdot - 3$ .

$\frac{y (- y - 3)}{y (y + 3) (y - 3)}$

Step 5.4

Cancel the common factor of $y$ .

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Step 5.4.1

Cancel the common factor.

$\frac{y (- y - 3)}{y (y + 3) (y - 3)}$

Step 5.4.2

Rewrite the expression.

$\frac{- y - 3}{(y + 3) (y - 3)}$

Step 5.5

Cancel the common factor of $- y - 3$ and $y + 3$ .

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Step 5.5.1

Factor $- 1$ out of $- y$ .

$\frac{- (y) - 3}{(y + 3) (y - 3)}$

Step 5.5.2

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{- (y) - 1 (3)}{(y + 3) (y - 3)}$

Step 5.5.3

Factor $- 1$ out of $- (y) - 1 (3)$ .

$\frac{- (y + 3)}{(y + 3) (y - 3)}$

Step 5.5.4

Rewrite $- (y + 3)$ as $- 1 (y + 3)$ .

$\frac{- 1 (y + 3)}{(y + 3) (y - 3)}$

Step 5.5.5

Cancel the common factor.

$\frac{- 1 (y + 3)}{(y + 3) (y - 3)}$

Step 5.5.6

Rewrite the expression.

$\frac{- 1}{y - 3}$

Step 5.6

Move the negative in front of the fraction.

$- \frac{1}{y - 3}$