Algebra Examples

$\frac{4}{y + 2} - \frac{3}{y - 2} + \frac{12}{y^{2} - 4}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $4$ as $2^{2}$ .

$\frac{4}{y + 2} - \frac{3}{y - 2} + \frac{12}{y^{2} - 2^{2}}$

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 $b = 2$ .

$\frac{4}{y + 2} - \frac{3}{y - 2} + \frac{12}{(y + 2) (y - 2)}$

Step 2

To write $\frac{4}{y + 2}$ as a fraction with a common denominator, multiply by $\frac{y - 2}{y - 2}$ .

$\frac{4}{y + 2} \cdot \frac{y - 2}{y - 2} - \frac{3}{y - 2} + \frac{12}{(y + 2) (y - 2)}$

Step 3

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

$\frac{4}{y + 2} \cdot \frac{y - 2}{y - 2} - \frac{3}{y - 2} \cdot \frac{y + 2}{y + 2} + \frac{12}{(y + 2) (y - 2)}$

Step 4

Write each expression with a common denominator of $(y + 2) (y - 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{4}{y + 2}$ by $\frac{y - 2}{y - 2}$ .

$\frac{4 (y - 2)}{(y + 2) (y - 2)} - \frac{3}{y - 2} \cdot \frac{y + 2}{y + 2} + \frac{12}{(y + 2) (y - 2)}$

Step 4.2

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

$\frac{4 (y - 2)}{(y + 2) (y - 2)} - \frac{3 (y + 2)}{(y - 2) (y + 2)} + \frac{12}{(y + 2) (y - 2)}$

Step 4.3

Reorder the factors of $(y - 2) (y + 2)$ .

$\frac{4 (y - 2)}{(y + 2) (y - 2)} - \frac{3 (y + 2)}{(y + 2) (y - 2)} + \frac{12}{(y + 2) (y - 2)}$

Step 5

Simplify terms.

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Step 5.1

Combine the numerators over the common denominator.

$\frac{4 (y - 2) - 3 (y + 2)}{(y + 2) (y - 2)} + \frac{12}{(y + 2) (y - 2)}$

Step 5.2

Combine the numerators over the common denominator.

$\frac{4 (y - 2) - 3 (y + 2) + 12}{(y + 2) (y - 2)}$

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

$\frac{4 y + 4 \cdot - 2 - 3 (y + 2) + 12}{(y + 2) (y - 2)}$

Step 6.2

Multiply $4$ by $- 2$ .

$\frac{4 y - 8 - 3 (y + 2) + 12}{(y + 2) (y - 2)}$

Step 6.3

Apply the distributive property.

$\frac{4 y - 8 - 3 y - 3 \cdot 2 + 12}{(y + 2) (y - 2)}$

Step 6.4

Multiply $- 3$ by $2$ .

$\frac{4 y - 8 - 3 y - 6 + 12}{(y + 2) (y - 2)}$

Step 7

Simplify terms.

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Step 7.1

Subtract $3 y$ from $4 y$ .

$\frac{y - 8 - 6 + 12}{(y + 2) (y - 2)}$

Step 7.2

Simplify by adding and subtracting.

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Step 7.2.1

Subtract $6$ from $- 8$ .

$\frac{y - 14 + 12}{(y + 2) (y - 2)}$

Step 7.2.2

Add $- 14$ and $12$ .

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

Step 7.3

Cancel the common factor of $y - 2$ .

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Step 7.3.1

Cancel the common factor.

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

Step 7.3.2

Rewrite the expression.

$\frac{1}{y + 2}$