Simplify (1-9w^2)/(12w-4)

$\frac{1 - 9 w^{2}}{12 w - 4}$

Step 1

Simplify the numerator.

Rewrite $1$ as $1^{2}$ .

$\frac{1^{2} - 9 w^{2}}{12 w - 4}$

Rewrite $9 w^{2}$ as ${(3 w)}^{2}$ .

$\frac{1^{2} - {(3 w)}^{2}}{12 w - 4}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = 3 w$ .

$\frac{(1 + 3 w) (1 - (3 w))}{12 w - 4}$

Multiply $3$ by $- 1$ .

$\frac{(1 + 3 w) (1 - 3 w)}{12 w - 4}$

Step 2

Simplify terms.

Factor $4$ out of $12 w - 4$ .

Factor $4$ out of $12 w$ .

$\frac{(1 + 3 w) (1 - 3 w)}{4 (3 w) - 4}$

Factor $4$ out of $- 4$ .

$\frac{(1 + 3 w) (1 - 3 w)}{4 (3 w) + 4 (- 1)}$

Factor $4$ out of $4 (3 w) + 4 (- 1)$ .

$\frac{(1 + 3 w) (1 - 3 w)}{4 (3 w - 1)}$

Cancel the common factor of $1 - 3 w$ and $3 w - 1$ .

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

$\frac{(1 + 3 w) (- 1 (- 1) - 3 w)}{4 (3 w - 1)}$

Factor $- 1$ out of $- 3 w$ .

$\frac{(1 + 3 w) (- 1 (- 1) - (3 w))}{4 (3 w - 1)}$

Factor $- 1$ out of $- 1 (- 1) - (3 w)$ .

$\frac{(1 + 3 w) (- 1 (- 1 + 3 w))}{4 (3 w - 1)}$

Reorder terms.

$\frac{(1 + 3 w) (- 1 (3 w - 1))}{4 (3 w - 1)}$

Cancel the common factor.

$\frac{(1 + 3 w) (- 1 (3 w - 1))}{4 (3 w - 1)}$

Rewrite the expression.

$\frac{(1 + 3 w) \cdot (- 1)}{4}$

Simplify the expression.

Move $- 1$ to the left of $1 + 3 w$ .

$\frac{- 1 \cdot (1 + 3 w)}{4}$

Move the negative in front of the fraction.

$- \frac{1 + 3 w}{4}$