Solve by Factoring 12/(y-6)-2/y=(16y+6)/(y^2-36)

$\frac{12}{y - 6} - \frac{2}{y} = \frac{16 y + 6}{y^{2} - 36}$

Step 1

Subtract $\frac{16 y + 6}{y^{2} - 36}$ from both sides of the equation.

$\frac{12}{y - 6} - \frac{2}{y} - \frac{16 y + 6}{y^{2} - 36} = 0$

Step 2

Simplify $\frac{12}{y - 6} - \frac{2}{y} - \frac{16 y + 6}{y^{2} - 36}$ .

Simplify each term.

Factor $2$ out of $16 y + 6$ .

Factor $2$ out of $16 y$ .

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

Factor $2$ out of $6$ .

$\frac{12}{y - 6} - \frac{2}{y} - \frac{2 (8 y) + 2 (3)}{y^{2} - 36} = 0$

Factor $2$ out of $2 (8 y) + 2 (3)$ .

$\frac{12}{y - 6} - \frac{2}{y} - \frac{2 (8 y + 3)}{y^{2} - 36} = 0$

Simplify the denominator.

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

$\frac{12}{y - 6} - \frac{2}{y} - \frac{2 (8 y + 3)}{y^{2} - 6^{2}} = 0$

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 $b = 6$ .

$\frac{12}{y - 6} - \frac{2}{y} - \frac{2 (8 y + 3)}{(y + 6) (y - 6)} = 0$

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

$\frac{12}{y - 6} \cdot \frac{y}{y} - \frac{2}{y} - \frac{2 (8 y + 3)}{(y + 6) (y - 6)} = 0$

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

$\frac{12}{y - 6} \cdot \frac{y}{y} - \frac{2}{y} \cdot \frac{y - 6}{y - 6} - \frac{2 (8 y + 3)}{(y + 6) (y - 6)} = 0$

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 $\frac{y}{y}$ .

$\frac{12 y}{(y - 6) y} - \frac{2}{y} \cdot \frac{y - 6}{y - 6} - \frac{2 (8 y + 3)}{(y + 6) (y - 6)} = 0$

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

$\frac{12 y}{(y - 6) y} - \frac{2 (y - 6)}{y (y - 6)} - \frac{2 (8 y + 3)}{(y + 6) (y - 6)} = 0$

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

$\frac{12 y}{y (y - 6)} - \frac{2 (y - 6)}{y (y - 6)} - \frac{2 (8 y + 3)}{(y + 6) (y - 6)} = 0$

Combine the numerators over the common denominator.

$\frac{12 y - 2 (y - 6)}{y (y - 6)} - \frac{2 (8 y + 3)}{(y + 6) (y - 6)} = 0$

Simplify the numerator.

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

Factor $2$ out of $12 y$ .

$\frac{2 (6 y) - 2 (y - 6)}{y (y - 6)} - \frac{2 (8 y + 3)}{(y + 6) (y - 6)} = 0$

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

$\frac{2 (6 y) + 2 (- (y - 6))}{y (y - 6)} - \frac{2 (8 y + 3)}{(y + 6) (y - 6)} = 0$

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

$\frac{2 (6 y - (y - 6))}{y (y - 6)} - \frac{2 (8 y + 3)}{(y + 6) (y - 6)} = 0$

Apply the distributive property.

$\frac{2 (6 y - y - - 6)}{y (y - 6)} - \frac{2 (8 y + 3)}{(y + 6) (y - 6)} = 0$

Multiply $- 1$ by $- 6$ .

$\frac{2 (6 y - y + 6)}{y (y - 6)} - \frac{2 (8 y + 3)}{(y + 6) (y - 6)} = 0$

Subtract $y$ from $6 y$ .

$\frac{2 (5 y + 6)}{y (y - 6)} - \frac{2 (8 y + 3)}{(y + 6) (y - 6)} = 0$

To write $\frac{2 (5 y + 6)}{y (y - 6)}$ as a fraction with a common denominator, multiply by $\frac{y + 6}{y + 6}$ .

$\frac{2 (5 y + 6)}{y (y - 6)} \cdot \frac{y + 6}{y + 6} - \frac{2 (8 y + 3)}{(y + 6) (y - 6)} = 0$

To write $- \frac{2 (8 y + 3)}{(y + 6) (y - 6)}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{2 (5 y + 6)}{y (y - 6)} \cdot \frac{y + 6}{y + 6} - \frac{2 (8 y + 3)}{(y + 6) (y - 6)} \cdot \frac{y}{y} = 0$

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

Multiply $\frac{2 (5 y + 6)}{y (y - 6)}$ by $\frac{y + 6}{y + 6}$ .

$\frac{2 (5 y + 6) (y + 6)}{y (y - 6) (y + 6)} - \frac{2 (8 y + 3)}{(y + 6) (y - 6)} \cdot \frac{y}{y} = 0$

Multiply $\frac{2 (8 y + 3)}{(y + 6) (y - 6)}$ by $\frac{y}{y}$ .

$\frac{2 (5 y + 6) (y + 6)}{y (y - 6) (y + 6)} - \frac{2 (8 y + 3) y}{(y + 6) (y - 6) y} = 0$

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

$\frac{2 (5 y + 6) (y + 6)}{y (y + 6) (y - 6)} - \frac{2 (8 y + 3) y}{(y + 6) (y - 6) y} = 0$

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

$\frac{2 (5 y + 6) (y + 6)}{y (y + 6) (y - 6)} - \frac{2 (8 y + 3) y}{y (y + 6) (y - 6)} = 0$

Combine the numerators over the common denominator.

$\frac{2 (5 y + 6) (y + 6) - 2 (8 y + 3) y}{y (y + 6) (y - 6)} = 0$

Simplify the numerator.

Factor $2$ out of $2 (5 y + 6) (y + 6) - 2 (8 y + 3) y$ .

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

$\frac{2 ((5 y + 6) (y + 6)) - 2 (8 y + 3) y}{y (y + 6) (y - 6)} = 0$

Factor $2$ out of $- 2 (8 y + 3) y$ .

$\frac{2 ((5 y + 6) (y + 6)) + 2 (- (8 y + 3) y)}{y (y + 6) (y - 6)} = 0$

Factor $2$ out of $2 ((5 y + 6) (y + 6)) + 2 (- (8 y + 3) y)$ .

$\frac{2 ((5 y + 6) (y + 6) - (8 y + 3) y)}{y (y + 6) (y - 6)} = 0$

Expand $(5 y + 6) (y + 6)$ using the FOIL Method.

Apply the distributive property.

$\frac{2 (5 y (y + 6) + 6 (y + 6) - (8 y + 3) y)}{y (y + 6) (y - 6)} = 0$

Apply the distributive property.

$\frac{2 (5 y \cdot y + 5 y \cdot 6 + 6 (y + 6) - (8 y + 3) y)}{y (y + 6) (y - 6)} = 0$

Apply the distributive property.

$\frac{2 (5 y \cdot y + 5 y \cdot 6 + 6 y + 6 \cdot 6 - (8 y + 3) y)}{y (y + 6) (y - 6)} = 0$

Simplify and combine like terms.

Simplify each term.

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

Move $y$ .

$\frac{2 (5 (y \cdot y) + 5 y \cdot 6 + 6 y + 6 \cdot 6 - (8 y + 3) y)}{y (y + 6) (y - 6)} = 0$

Multiply $y$ by $y$ .

$\frac{2 (5 y^{2} + 5 y \cdot 6 + 6 y + 6 \cdot 6 - (8 y + 3) y)}{y (y + 6) (y - 6)} = 0$

Multiply $6$ by $5$ .

$\frac{2 (5 y^{2} + 30 y + 6 y + 6 \cdot 6 - (8 y + 3) y)}{y (y + 6) (y - 6)} = 0$

Multiply $6$ by $6$ .

$\frac{2 (5 y^{2} + 30 y + 6 y + 36 - (8 y + 3) y)}{y (y + 6) (y - 6)} = 0$

Add $30 y$ and $6 y$ .

$\frac{2 (5 y^{2} + 36 y + 36 - (8 y + 3) y)}{y (y + 6) (y - 6)} = 0$

Apply the distributive property.

$\frac{2 (5 y^{2} + 36 y + 36 + (- (8 y) - 1 \cdot 3) y)}{y (y + 6) (y - 6)} = 0$

Multiply $8$ by $- 1$ .

$\frac{2 (5 y^{2} + 36 y + 36 + (- 8 y - 1 \cdot 3) y)}{y (y + 6) (y - 6)} = 0$

Multiply $- 1$ by $3$ .

$\frac{2 (5 y^{2} + 36 y + 36 + (- 8 y - 3) y)}{y (y + 6) (y - 6)} = 0$

Apply the distributive property.

$\frac{2 (5 y^{2} + 36 y + 36 - 8 y \cdot y - 3 y)}{y (y + 6) (y - 6)} = 0$

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

Move $y$ .

$\frac{2 (5 y^{2} + 36 y + 36 - 8 (y \cdot y) - 3 y)}{y (y + 6) (y - 6)} = 0$

Multiply $y$ by $y$ .

$\frac{2 (5 y^{2} + 36 y + 36 - 8 y^{2} - 3 y)}{y (y + 6) (y - 6)} = 0$

Subtract $8 y^{2}$ from $5 y^{2}$ .

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

Subtract $3 y$ from $36 y$ .

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

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 $36$ .

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

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

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

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

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

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 12 = - 12$ and whose sum is $b = 11$ .

Factor $11$ out of $11 y$ .

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

Rewrite $11$ as $- 1$ plus $12$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

Factor out the greatest common factor (GCF) from each group.

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

Factor the polynomial by factoring out the greatest common factor, $- y - 1$ .

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

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

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

Multiply $2$ by $3$ .

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

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

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

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

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

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

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

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

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

Move the negative in front of the fraction.

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

Reorder factors in $- \frac{(6 (y + 1)) (y - 12)}{y (y + 6) (y - 6)}$ .

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

Step 3

Set the numerator equal to zero.

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

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 $0$ .

$y + 1 = 0$

$y - 12 = 0$

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.

$y = - 1$

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.

$y = 12$

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

$y = - 1, 12$

Solve by Factoring 12/(y-6)-2/y=(16y+6)/(y^2-36)

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