Simplify (x^2-16)/(x^2-8x+16)

$\frac{x^{2} - 16}{x^{2} - 8 x + 16}$

Step 1

Simplify the numerator.

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

$\frac{x^{2} - 4^{2}}{x^{2} - 8 x + 16}$

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 $b = 4$ .

$\frac{(x + 4) (x - 4)}{x^{2} - 8 x + 16}$

Step 2

Factor using the perfect square rule.

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

$\frac{(x + 4) (x - 4)}{x^{2} - 8 x + 4^{2}}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 x = 2 \cdot x \cdot 4$

Rewrite the polynomial.

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

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 4$ .

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

Step 3

Cancel the common factor of $x - 4$ and ${(x - 4)}^{2}$ .

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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