Simplify (x^2-36)/(42x-7x^2)

$\frac{x^{2} - 36}{42 x - 7 x^{2}}$

Step 1

Simplify the numerator.

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

$\frac{x^{2} - 6^{2}}{42 x - 7 x^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 6$ .

$\frac{(x + 6) (x - 6)}{42 x - 7 x^{2}}$

Step 2

Simplify terms.

Factor $7 x$ out of $42 x - 7 x^{2}$ .

Factor $7 x$ out of $42 x$ .

$\frac{(x + 6) (x - 6)}{7 x (6) - 7 x^{2}}$

Factor $7 x$ out of $- 7 x^{2}$ .

$\frac{(x + 6) (x - 6)}{7 x (6) + 7 x (- x)}$

Factor $7 x$ out of $7 x (6) + 7 x (- x)$ .

$\frac{(x + 6) (x - 6)}{7 x (6 - x)}$

Cancel the common factor of $x - 6$ and $6 - x$ .

Factor $- 1$ out of $x$ .

$\frac{(x + 6) (- 1 (- x) - 6)}{7 x (6 - x)}$

Rewrite $- 6$ as $- 1 (6)$ .

$\frac{(x + 6) (- 1 (- x) - 1 (6))}{7 x (6 - x)}$

Factor $- 1$ out of $- 1 (- x) - 1 (6)$ .

$\frac{(x + 6) (- 1 (- x + 6))}{7 x (6 - x)}$

Reorder terms.

$\frac{(x + 6) (- 1 (- x + 6))}{7 x (- x + 6)}$

Cancel the common factor.

$\frac{(x + 6) (- 1 (- x + 6))}{7 x (- x + 6)}$

Rewrite the expression.

$\frac{(x + 6) \cdot (- 1)}{7 x}$

Simplify the expression.

Move $- 1$ to the left of $x + 6$ .

$\frac{- 1 \cdot (x + 6)}{7 x}$

Move the negative in front of the fraction.

$- \frac{x + 6}{7 x}$