Solve by Factoring 12/(y+6)+2/y=(8y-6)/(y^2-36)

$\frac{12}{y + 6} + \frac{2}{y} = \frac{8 y - 6}{y^{2} - 36}$

Step 1

Subtract $\frac{8 y - 6}{y^{2} - 36}$ from both sides of the equation.

$\frac{12}{y + 6} + \frac{2}{y} - \frac{8 y - 6}{y^{2} - 36} = 0$

Step 2

Simplify $\frac{12}{y + 6} + \frac{2}{y} - \frac{8 y - 6}{y^{2} - 36}$ .

Simplify each term.

Factor $2$ out of $8 y - 6$ .

Factor $2$ out of $8 y$ .

$\frac{12}{y + 6} + \frac{2}{y} - \frac{2 (4 y) - 6}{y^{2} - 36} = 0$

Factor $2$ out of $- 6$ .

$\frac{12}{y + 6} + \frac{2}{y} - \frac{2 (4 y) + 2 (- 3)}{y^{2} - 36} = 0$

Factor $2$ out of $2 (4 y) + 2 (- 3)$ .

$\frac{12}{y + 6} + \frac{2}{y} - \frac{2 (4 y - 3)}{y^{2} - 36} = 0$

Simplify the denominator.

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

$\frac{12}{y + 6} + \frac{2}{y} - \frac{2 (4 y - 3)}{y^{2} - 6^{2}} = 0$

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 $b = 6$ .

$\frac{12}{y + 6} + \frac{2}{y} - \frac{2 (4 y - 3)}{(y + 6) (y - 6)} = 0$

To write $\frac{12}{y + 6}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{12}{y + 6} \cdot \frac{y}{y} + \frac{2}{y} - \frac{2 (4 y - 3)}{(y + 6) (y - 6)} = 0$

To write $\frac{2}{y}$ as a fraction with a common denominator, multiply by $\frac{y + 6}{y + 6}$ .

$\frac{12}{y + 6} \cdot \frac{y}{y} + \frac{2}{y} \cdot \frac{y + 6}{y + 6} - \frac{2 (4 y - 3)}{(y + 6) (y - 6)} = 0$

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

Multiply $\frac{12}{y + 6}$ by $\frac{y}{y}$ .

$\frac{12 y}{(y + 6) y} + \frac{2}{y} \cdot \frac{y + 6}{y + 6} - \frac{2 (4 y - 3)}{(y + 6) (y - 6)} = 0$

Multiply $\frac{2}{y}$ by $\frac{y + 6}{y + 6}$ .

$\frac{12 y}{(y + 6) y} + \frac{2 (y + 6)}{y (y + 6)} - \frac{2 (4 y - 3)}{(y + 6) (y - 6)} = 0$

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

$\frac{12 y}{y (y + 6)} + \frac{2 (y + 6)}{y (y + 6)} - \frac{2 (4 y - 3)}{(y + 6) (y - 6)} = 0$

Combine the numerators over the common denominator.

$\frac{12 y + 2 (y + 6)}{y (y + 6)} - \frac{2 (4 y - 3)}{(y + 6) (y - 6)} = 0$

Simplify the numerator.

Factor $2$ out of $12 y + 2 (y + 6)$ .

Factor $2$ out of $12 y$ .

$\frac{2 (6 y) + 2 (y + 6)}{y (y + 6)} - \frac{2 (4 y - 3)}{(y + 6) (y - 6)} = 0$

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

$\frac{2 (6 y + y + 6)}{y (y + 6)} - \frac{2 (4 y - 3)}{(y + 6) (y - 6)} = 0$

Add $6 y$ and $y$ .

$\frac{2 (7 y + 6)}{y (y + 6)} - \frac{2 (4 y - 3)}{(y + 6) (y - 6)} = 0$

To write $\frac{2 (7 y + 6)}{y (y + 6)}$ as a fraction with a common denominator, multiply by $\frac{y - 6}{y - 6}$ .

$\frac{2 (7 y + 6)}{y (y + 6)} \cdot \frac{y - 6}{y - 6} - \frac{2 (4 y - 3)}{(y + 6) (y - 6)} = 0$

To write $- \frac{2 (4 y - 3)}{(y + 6) (y - 6)}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{2 (7 y + 6)}{y (y + 6)} \cdot \frac{y - 6}{y - 6} - \frac{2 (4 y - 3)}{(y + 6) (y - 6)} \cdot \frac{y}{y} = 0$

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

Multiply $\frac{2 (7 y + 6)}{y (y + 6)}$ by $\frac{y - 6}{y - 6}$ .

$\frac{2 (7 y + 6) (y - 6)}{y (y + 6) (y - 6)} - \frac{2 (4 y - 3)}{(y + 6) (y - 6)} \cdot \frac{y}{y} = 0$

Multiply $\frac{2 (4 y - 3)}{(y + 6) (y - 6)}$ by $\frac{y}{y}$ .

$\frac{2 (7 y + 6) (y - 6)}{y (y + 6) (y - 6)} - \frac{2 (4 y - 3) y}{(y + 6) (y - 6) y} = 0$

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

$\frac{2 (7 y + 6) (y - 6)}{y (y + 6) (y - 6)} - \frac{2 (4 y - 3) y}{y (y + 6) (y - 6)} = 0$

Combine the numerators over the common denominator.

$\frac{2 (7 y + 6) (y - 6) - 2 (4 y - 3) y}{y (y + 6) (y - 6)} = 0$

Simplify the numerator.

Factor $2$ out of $2 (7 y + 6) (y - 6) - 2 (4 y - 3) y$ .

Factor $2$ out of $2 (7 y + 6) (y - 6)$ .

$\frac{2 ((7 y + 6) (y - 6)) - 2 (4 y - 3) y}{y (y + 6) (y - 6)} = 0$

Factor $2$ out of $- 2 (4 y - 3) y$ .

$\frac{2 ((7 y + 6) (y - 6)) + 2 (- (4 y - 3) y)}{y (y + 6) (y - 6)} = 0$

Factor $2$ out of $2 ((7 y + 6) (y - 6)) + 2 (- (4 y - 3) y)$ .

$\frac{2 ((7 y + 6) (y - 6) - (4 y - 3) y)}{y (y + 6) (y - 6)} = 0$

Expand $(7 y + 6) (y - 6)$ using the FOIL Method.

Apply the distributive property.

$\frac{2 (7 y (y - 6) + 6 (y - 6) - (4 y - 3) y)}{y (y + 6) (y - 6)} = 0$

Apply the distributive property.

$\frac{2 (7 y \cdot y + 7 y \cdot - 6 + 6 (y - 6) - (4 y - 3) y)}{y (y + 6) (y - 6)} = 0$

Apply the distributive property.

$\frac{2 (7 y \cdot y + 7 y \cdot - 6 + 6 y + 6 \cdot - 6 - (4 y - 3) y)}{y (y + 6) (y - 6)} = 0$

Simplify and combine like terms.

Simplify each term.

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

Move $y$ .

$\frac{2 (7 (y \cdot y) + 7 y \cdot - 6 + 6 y + 6 \cdot - 6 - (4 y - 3) y)}{y (y + 6) (y - 6)} = 0$

Multiply $y$ by $y$ .

$\frac{2 (7 y^{2} + 7 y \cdot - 6 + 6 y + 6 \cdot - 6 - (4 y - 3) y)}{y (y + 6) (y - 6)} = 0$

Multiply $- 6$ by $7$ .

$\frac{2 (7 y^{2} - 42 y + 6 y + 6 \cdot - 6 - (4 y - 3) y)}{y (y + 6) (y - 6)} = 0$

Multiply $6$ by $- 6$ .

$\frac{2 (7 y^{2} - 42 y + 6 y - 36 - (4 y - 3) y)}{y (y + 6) (y - 6)} = 0$

Add $- 42 y$ and $6 y$ .

$\frac{2 (7 y^{2} - 36 y - 36 - (4 y - 3) y)}{y (y + 6) (y - 6)} = 0$

Apply the distributive property.

$\frac{2 (7 y^{2} - 36 y - 36 + (- (4 y) - - 3) y)}{y (y + 6) (y - 6)} = 0$

Multiply $4$ by $- 1$ .

$\frac{2 (7 y^{2} - 36 y - 36 + (- 4 y - - 3) y)}{y (y + 6) (y - 6)} = 0$

Multiply $- 1$ by $- 3$ .

$\frac{2 (7 y^{2} - 36 y - 36 + (- 4 y + 3) y)}{y (y + 6) (y - 6)} = 0$

Apply the distributive property.

$\frac{2 (7 y^{2} - 36 y - 36 - 4 y \cdot y + 3 y)}{y (y + 6) (y - 6)} = 0$

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

Move $y$ .

$\frac{2 (7 y^{2} - 36 y - 36 - 4 (y \cdot y) + 3 y)}{y (y + 6) (y - 6)} = 0$

Multiply $y$ by $y$ .

$\frac{2 (7 y^{2} - 36 y - 36 - 4 y^{2} + 3 y)}{y (y + 6) (y - 6)} = 0$

Subtract $4 y^{2}$ from $7 y^{2}$ .

$\frac{2 (3 y^{2} - 36 y - 36 + 3 y)}{y (y + 6) (y - 6)} = 0$

Add $- 36 y$ and $3 y$ .

$\frac{2 (3 y^{2} - 33 y - 36)}{y (y + 6) (y - 6)} = 0$

Rewrite $3 y^{2} - 33 y - 36$ in a factored form.

Factor $3$ out of $3 y^{2} - 33 y - 36$ .

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

$\frac{2 (3 (y^{2}) - 33 y - 36)}{y (y + 6) (y - 6)} = 0$

Factor $3$ out of $- 33 y$ .

$\frac{2 (3 (y^{2}) + 3 (- 11 y) - 36)}{y (y + 6) (y - 6)} = 0$

Factor $3$ out of $- 36$ .

$\frac{2 (3 y^{2} + 3 (- 11 y) + 3 \cdot - 12)}{y (y + 6) (y - 6)} = 0$

Factor $3$ out of $3 y^{2} + 3 (- 11 y)$ .

$\frac{2 (3 (y^{2} - 11 y) + 3 \cdot - 12)}{y (y + 6) (y - 6)} = 0$

Factor $3$ out of $3 (y^{2} - 11 y) + 3 \cdot - 12$ .

$\frac{2 (3 (y^{2} - 11 y - 12))}{y (y + 6) (y - 6)} = 0$

Factor $y^{2} - 11 y - 12$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $- 11$ .

$- 12, 1$

Write the factored form using these integers.

$\frac{2 (3 ((y - 12) (y + 1)))}{y (y + 6) (y - 6)} = 0$

$\frac{2 (3 (y - 12) (y + 1))}{y (y + 6) (y - 6)} = 0$

$\frac{2 \cdot 3 (y - 12) (y + 1)}{y (y + 6) (y - 6)} = 0$

Multiply $2$ by $3$ .

$\frac{6 (y - 12) (y + 1)}{y (y + 6) (y - 6)} = 0$

Step 3

Set the numerator equal to zero.

$6 (y - 12) (y + 1) = 0$

Step 4

Solve the equation for $y$ .

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y - 12 = 0$

$y + 1 = 0$

Set $y - 12$ equal to $0$ and solve for $y$ .

Set $y - 12$ equal to $0$ .

$y - 12 = 0$

Add $12$ to both sides of the equation.

$y = 12$

Set $y + 1$ equal to $0$ and solve for $y$ .

Set $y + 1$ equal to $0$ .

$y + 1 = 0$

Subtract $1$ from both sides of the equation.

$y = - 1$

The final solution is all the values that make $6 (y - 12) (y + 1) = 0$ true.

$y = 12, - 1$

Solve by Factoring 12/(y+6)+2/y=(8y-6)/(y^2-36)

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