Algebra Examples

$\frac{5 y}{y + 6} + \frac{7 y^{2}}{y^{2} - 36} - \frac{4}{y - 6}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $36$ as $6^{2}$ .

$\frac{5 y}{y + 6} + \frac{7 y^{2}}{y^{2} - 6^{2}} - \frac{4}{y - 6}$

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 $b = 6$ .

$\frac{5 y}{y + 6} + \frac{7 y^{2}}{(y + 6) (y - 6)} - \frac{4}{y - 6}$

Step 2

To write $\frac{5 y}{y + 6}$ as a fraction with a common denominator, multiply by $\frac{y - 6}{y - 6}$ .

$\frac{5 y}{y + 6} \cdot \frac{y - 6}{y - 6} + \frac{7 y^{2}}{(y + 6) (y - 6)} - \frac{4}{y - 6}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{5 y}{y + 6}$ by $\frac{y - 6}{y - 6}$ .

$\frac{5 y (y - 6)}{(y + 6) (y - 6)} + \frac{7 y^{2}}{(y + 6) (y - 6)} - \frac{4}{y - 6}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{5 y (y - 6) + 7 y^{2}}{(y + 6) (y - 6)} - \frac{4}{y - 6}$

Step 4

Simplify each term.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor $y$ out of $5 y (y - 6) + 7 y^{2}$ .

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Step 4.1.1.1

Factor $y$ out of $5 y (y - 6)$ .

$\frac{y (5 (y - 6)) + 7 y^{2}}{(y + 6) (y - 6)} - \frac{4}{y - 6}$

Step 4.1.1.2

Factor $y$ out of $7 y^{2}$ .

$\frac{y (5 (y - 6)) + y (7 y)}{(y + 6) (y - 6)} - \frac{4}{y - 6}$

Step 4.1.1.3

Factor $y$ out of $y (5 (y - 6)) + y (7 y)$ .

$\frac{y (5 (y - 6) + 7 y)}{(y + 6) (y - 6)} - \frac{4}{y - 6}$

Step 4.1.2

Apply the distributive property.

$\frac{y (5 y + 5 \cdot - 6 + 7 y)}{(y + 6) (y - 6)} - \frac{4}{y - 6}$

Step 4.1.3

Multiply $5$ by $- 6$ .

$\frac{y (5 y - 30 + 7 y)}{(y + 6) (y - 6)} - \frac{4}{y - 6}$

Step 4.1.4

Add $5 y$ and $7 y$ .

$\frac{y (12 y - 30)}{(y + 6) (y - 6)} - \frac{4}{y - 6}$

Step 4.1.5

Factor $6$ out of $12 y - 30$ .

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Step 4.1.5.1

Factor $6$ out of $12 y$ .

$\frac{y (6 (2 y) - 30)}{(y + 6) (y - 6)} - \frac{4}{y - 6}$

Step 4.1.5.2

Factor $6$ out of $- 30$ .

$\frac{y (6 (2 y) + 6 (- 5))}{(y + 6) (y - 6)} - \frac{4}{y - 6}$

Step 4.1.5.3

Factor $6$ out of $6 (2 y) + 6 (- 5)$ .

$\frac{y (6 (2 y - 5))}{(y + 6) (y - 6)} - \frac{4}{y - 6}$

$\frac{y \cdot 6 (2 y - 5)}{(y + 6) (y - 6)} - \frac{4}{y - 6}$

Step 4.2

Move $6$ to the left of $y$ .

$\frac{6 y (2 y - 5)}{(y + 6) (y - 6)} - \frac{4}{y - 6}$

Step 5

To write $- \frac{4}{y - 6}$ as a fraction with a common denominator, multiply by $\frac{y + 6}{y + 6}$ .

$\frac{6 y (2 y - 5)}{(y + 6) (y - 6)} - \frac{4}{y - 6} \cdot \frac{y + 6}{y + 6}$

Step 6

Write each expression with a common denominator of $(y + 6) (y - 6)$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1

Multiply $\frac{4}{y - 6}$ by $\frac{y + 6}{y + 6}$ .

$\frac{6 y (2 y - 5)}{(y + 6) (y - 6)} - \frac{4 (y + 6)}{(y - 6) (y + 6)}$

Step 6.2

Reorder the factors of $(y - 6) (y + 6)$ .

$\frac{6 y (2 y - 5)}{(y + 6) (y - 6)} - \frac{4 (y + 6)}{(y + 6) (y - 6)}$

Step 7

Combine the numerators over the common denominator.

$\frac{6 y (2 y - 5) - 4 (y + 6)}{(y + 6) (y - 6)}$

Step 8

Simplify the numerator.

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Step 8.1

Factor $2$ out of $6 y (2 y - 5) - 4 (y + 6)$ .

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Step 8.1.1

Factor $2$ out of $6 y (2 y - 5)$ .

$\frac{2 (3 y (2 y - 5)) - 4 (y + 6)}{(y + 6) (y - 6)}$

Step 8.1.2

Factor $2$ out of $- 4 (y + 6)$ .

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

Step 8.1.3

Factor $2$ out of $2 (3 y (2 y - 5)) + 2 (- 2 (y + 6))$ .

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

Step 8.2

Apply the distributive property.

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

Step 8.3

Rewrite using the commutative property of multiplication.

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

Step 8.4

Multiply $- 5$ by $3$ .

$\frac{2 (3 \cdot 2 y \cdot y - 15 y - 2 (y + 6))}{(y + 6) (y - 6)}$

Step 8.5

Simplify each term.

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Step 8.5.1

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

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Step 8.5.1.1

Move $y$ .

$\frac{2 (3 \cdot 2 (y \cdot y) - 15 y - 2 (y + 6))}{(y + 6) (y - 6)}$

Step 8.5.1.2

Multiply $y$ by $y$ .

$\frac{2 (3 \cdot 2 y^{2} - 15 y - 2 (y + 6))}{(y + 6) (y - 6)}$

Step 8.5.2

Multiply $3$ by $2$ .

$\frac{2 (6 y^{2} - 15 y - 2 (y + 6))}{(y + 6) (y - 6)}$

Step 8.6

Apply the distributive property.

$\frac{2 (6 y^{2} - 15 y - 2 y - 2 \cdot 6)}{(y + 6) (y - 6)}$

Step 8.7

Multiply $- 2$ by $6$ .

$\frac{2 (6 y^{2} - 15 y - 2 y - 12)}{(y + 6) (y - 6)}$

Step 8.8

Subtract $2 y$ from $- 15 y$ .

$\frac{2 (6 y^{2} - 17 y - 12)}{(y + 6) (y - 6)}$