Solve by Factoring 12/(v^2-16)-24/(v-4)=3

$\frac{12}{v^{2} - 16} - \frac{24}{v - 4} = 3$

Step 1

Subtract $3$ from both sides of the equation.

$\frac{12}{v^{2} - 16} - \frac{24}{v - 4} - 3 = 0$

Step 2

Simplify $\frac{12}{v^{2} - 16} - \frac{24}{v - 4} - 3$ .

Simplify the denominator.

Rewrite $16$ as $4^{2}$ .

$\frac{12}{v^{2} - 4^{2}} - \frac{24}{v - 4} - 3 = 0$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = v$ and $b = 4$ .

$\frac{12}{(v + 4) (v - 4)} - \frac{24}{v - 4} - 3 = 0$

To write $- \frac{24}{v - 4}$ as a fraction with a common denominator, multiply by $\frac{v + 4}{v + 4}$ .

$\frac{12}{(v + 4) (v - 4)} - \frac{24}{v - 4} \cdot \frac{v + 4}{v + 4} - 3 = 0$

Write each expression with a common denominator of $(v + 4) (v - 4)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{24}{v - 4}$ by $\frac{v + 4}{v + 4}$ .

$\frac{12}{(v + 4) (v - 4)} - \frac{24 (v + 4)}{(v - 4) (v + 4)} - 3 = 0$

Reorder the factors of $(v - 4) (v + 4)$ .

$\frac{12}{(v + 4) (v - 4)} - \frac{24 (v + 4)}{(v + 4) (v - 4)} - 3 = 0$

Combine the numerators over the common denominator.

$\frac{12 - 24 (v + 4)}{(v + 4) (v - 4)} - 3 = 0$

Simplify the numerator.

Factor $12$ out of $12 - 24 (v + 4)$ .

Factor $12$ out of $12$ .

$\frac{12 (1) - 24 (v + 4)}{(v + 4) (v - 4)} - 3 = 0$

Factor $12$ out of $- 24 (v + 4)$ .

$\frac{12 (1) + 12 (- 2 (v + 4))}{(v + 4) (v - 4)} - 3 = 0$

Factor $12$ out of $12 (1) + 12 (- 2 (v + 4))$ .

$\frac{12 (1 - 2 (v + 4))}{(v + 4) (v - 4)} - 3 = 0$

Apply the distributive property.

$\frac{12 (1 - 2 v - 2 \cdot 4)}{(v + 4) (v - 4)} - 3 = 0$

Multiply $- 2$ by $4$ .

$\frac{12 (1 - 2 v - 8)}{(v + 4) (v - 4)} - 3 = 0$

Subtract $8$ from $1$ .

$\frac{12 (- 2 v - 7)}{(v + 4) (v - 4)} - 3 = 0$

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

$\frac{12 (- 2 v - 7)}{(v + 4) (v - 4)} - 3 \cdot \frac{(v + 4) (v - 4)}{(v + 4) (v - 4)} = 0$

Combine $- 3$ and $\frac{(v + 4) (v - 4)}{(v + 4) (v - 4)}$ .

$\frac{12 (- 2 v - 7)}{(v + 4) (v - 4)} + \frac{- 3 ((v + 4) (v - 4))}{(v + 4) (v - 4)} = 0$

Combine the numerators over the common denominator.

$\frac{12 (- 2 v - 7) - 3 ((v + 4) (v - 4))}{(v + 4) (v - 4)} = 0$

Simplify the numerator.

Factor $3$ out of $12 (- 2 v - 7) - 3 (v + 4) (v - 4)$ .

Factor $3$ out of $12 (- 2 v - 7)$ .

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

Factor $3$ out of $- 3 (v + 4) (v - 4)$ .

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

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

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

Apply the distributive property.

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

Multiply $- 2$ by $4$ .

$\frac{3 (- 8 v + 4 \cdot - 7 - (v + 4) (v - 4))}{(v + 4) (v - 4)} = 0$

Multiply $4$ by $- 7$ .

$\frac{3 (- 8 v - 28 - (v + 4) (v - 4))}{(v + 4) (v - 4)} = 0$

Apply the distributive property.

$\frac{3 (- 8 v - 28 + (- v - 1 \cdot 4) (v - 4))}{(v + 4) (v - 4)} = 0$

Multiply $- 1$ by $4$ .

$\frac{3 (- 8 v - 28 + (- v - 4) (v - 4))}{(v + 4) (v - 4)} = 0$

Expand $(- v - 4) (v - 4)$ using the FOIL Method.

Apply the distributive property.

$\frac{3 (- 8 v - 28 - v (v - 4) - 4 (v - 4))}{(v + 4) (v - 4)} = 0$

Apply the distributive property.

$\frac{3 (- 8 v - 28 - v \cdot v - v \cdot - 4 - 4 (v - 4))}{(v + 4) (v - 4)} = 0$

Apply the distributive property.

$\frac{3 (- 8 v - 28 - v \cdot v - v \cdot - 4 - 4 v - 4 \cdot - 4)}{(v + 4) (v - 4)} = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $v$ by $v$ by adding the exponents.

Move $v$ .

$\frac{3 (- 8 v - 28 - (v \cdot v) - v \cdot - 4 - 4 v - 4 \cdot - 4)}{(v + 4) (v - 4)} = 0$

Multiply $v$ by $v$ .

$\frac{3 (- 8 v - 28 - v^{2} - v \cdot - 4 - 4 v - 4 \cdot - 4)}{(v + 4) (v - 4)} = 0$

Multiply $- 4$ by $- 1$ .

$\frac{3 (- 8 v - 28 - v^{2} + 4 v - 4 v - 4 \cdot - 4)}{(v + 4) (v - 4)} = 0$

Multiply $- 4$ by $- 4$ .

$\frac{3 (- 8 v - 28 - v^{2} + 4 v - 4 v + 16)}{(v + 4) (v - 4)} = 0$

Subtract $4 v$ from $4 v$ .

$\frac{3 (- 8 v - 28 - v^{2} + 0 + 16)}{(v + 4) (v - 4)} = 0$

Add $- v^{2}$ and $0$ .

$\frac{3 (- 8 v - 28 - v^{2} + 16)}{(v + 4) (v - 4)} = 0$

Add $- 28$ and $16$ .

$\frac{3 (- 8 v - v^{2} - 12)}{(v + 4) (v - 4)} = 0$

Reorder terms.

$\frac{3 (- v^{2} - 8 v - 12)}{(v + 4) (v - 4)} = 0$

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 $b = - 8$ .

Factor $- 8$ out of $- 8 v$ .

$\frac{3 (- v^{2} - 8 (v) - 12)}{(v + 4) (v - 4)} = 0$

Rewrite $- 8$ as $- 2$ plus $- 6$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

Factor the polynomial by factoring out the greatest common factor, $- v - 2$ .

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

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

Factor $- 1$ out of $- v$ .

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

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{3 (- (v) - 1 (2)) (v + 6)}{(v + 4) (v - 4)} = 0$

Factor $- 1$ out of $- (v) - 1 (2)$ .

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

Rewrite $- (v + 2)$ as $- 1 (v + 2)$ .

$\frac{3 (- 1 (v + 2)) (v + 6)}{(v + 4) (v - 4)} = 0$

Move the negative in front of the fraction.

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

Reorder factors in $- \frac{(3 (v + 2)) (v + 6)}{(v + 4) (v - 4)}$ .

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

Step 3

Set the numerator equal to zero.

$3 (v + 2) (v + 6) = 0$

Step 4

Solve the equation for $v$ .

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

$v + 2 = 0$

$v + 6 = 0$

Set $v + 2$ equal to $0$ and solve for $v$ .

Set $v + 2$ equal to $0$ .

$v + 2 = 0$

Subtract $2$ from both sides of the equation.

$v = - 2$

Set $v + 6$ equal to $0$ and solve for $v$ .

Set $v + 6$ equal to $0$ .

$v + 6 = 0$

Subtract $6$ from both sides of the equation.

$v = - 6$

The final solution is all the values that make $3 (v + 2) (v + 6) = 0$ true.

$v = - 2, - 6$

Solve by Factoring 12/(v^2-16)-24/(v-4)=3

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