Simplify (x^2-4x)/(16-x^2)

$\frac{x^{2} - 4 x}{16 - x^{2}}$

Step 1

Factor $x$ out of $x^{2} - 4 x$ .

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

$\frac{x \cdot x - 4 x}{16 - x^{2}}$

Factor $x$ out of $- 4 x$ .

$\frac{x \cdot x + x \cdot - 4}{16 - x^{2}}$

Factor $x$ out of $x \cdot x + x \cdot - 4$ .

$\frac{x (x - 4)}{16 - x^{2}}$

Step 2

Simplify the denominator.

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

$\frac{x (x - 4)}{4^{2} - x^{2}}$

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

$\frac{x (x - 4)}{(4 + x) (4 - x)}$

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x - 4$ and $4 - x$ .

Factor $- 1$ out of $x$ .

$\frac{x (- 1 (- x) - 4)}{(4 + x) (4 - x)}$

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{x (- 1 (- x) - 1 (4))}{(4 + x) (4 - x)}$

Factor $- 1$ out of $- 1 (- x) - 1 (4)$ .

$\frac{x (- 1 (- x + 4))}{(4 + x) (4 - x)}$

Reorder terms.

$\frac{x (- 1 (- x + 4))}{(4 + x) (- x + 4)}$

Cancel the common factor.

$\frac{x (- 1 (- x + 4))}{(4 + x) (- x + 4)}$

Rewrite the expression.

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

Simplify the expression.

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

$\frac{- 1 \cdot x}{4 + x}$

Move the negative in front of the fraction.

$- \frac{x}{4 + x}$