Basic Math Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

Step 1.2

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 $b = 3$ .

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

Step 2

To write $\frac{2 y}{y + 3}$ as a fraction with a common denominator, multiply by $\frac{y - 3}{y - 3}$ .

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

Step 3

To write $\frac{3 y}{y - 3}$ as a fraction with a common denominator, multiply by $\frac{y + 3}{y + 3}$ .

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

Step 4

Write each expression with a common denominator of $(y + 3) (y - 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2 y}{y + 3}$ by $\frac{y - 3}{y - 3}$ .

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Simplify terms.

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Step 5.1

Combine the numerators over the common denominator.

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

Step 5.2

Combine the numerators over the common denominator.

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

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

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

Step 6.2

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

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Step 6.2.1

Move $y$ .

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

Step 6.2.2

Multiply $y$ by $y$ .

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

Step 6.3

Multiply $- 3$ by $2$ .

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

Step 6.4

Apply the distributive property.

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

Step 6.5

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

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Step 6.5.1

Move $y$ .

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

Step 6.5.2

Multiply $y$ by $y$ .

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

Step 6.6

Multiply $3$ by $3$ .

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

Step 7

Simplify by adding terms.

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Step 7.1

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

$\frac{5 y^{2} - 6 y + 9 y - y^{2} + 3}{(y + 3) (y - 3)}$

Step 7.2

Subtract $y^{2}$ from $5 y^{2}$ .

$\frac{4 y^{2} - 6 y + 9 y + 3}{(y + 3) (y - 3)}$

Step 7.3

Add $- 6 y$ and $9 y$ .

$\frac{4 y^{2} + 3 y + 3}{(y + 3) (y - 3)}$