Simplify (x^2-9)/(x^2-6x+9)

$\frac{x^{2} - 9}{x^{2} - 6 x + 9}$

Step 1

Simplify the numerator.

Rewrite $9$ as $3^{2}$ .

$\frac{x^{2} - 3^{2}}{x^{2} - 6 x + 9}$

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 $b = 3$ .

$\frac{(x + 3) (x - 3)}{x^{2} - 6 x + 9}$

Step 2

Factor using the perfect square rule.

Rewrite $9$ as $3^{2}$ .

$\frac{(x + 3) (x - 3)}{x^{2} - 6 x + 3^{2}}$

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

$6 x = 2 \cdot x \cdot 3$

Rewrite the polynomial.

$\frac{(x + 3) (x - 3)}{x^{2} - 2 \cdot x \cdot 3 + 3^{2}}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 3$ .

$\frac{(x + 3) (x - 3)}{{(x - 3)}^{2}}$

Step 3

Cancel the common factor of $x - 3$ and ${(x - 3)}^{2}$ .

Factor $x - 3$ out of $(x + 3) (x - 3)$ .

$\frac{(x - 3) (x + 3)}{{(x - 3)}^{2}}$

Cancel the common factors.

Factor $x - 3$ out of ${(x - 3)}^{2}$ .

$\frac{(x - 3) (x + 3)}{(x - 3) (x - 3)}$

Cancel the common factor.

$\frac{(x - 3) (x + 3)}{(x - 3) (x - 3)}$

Rewrite the expression.

$\frac{x + 3}{x - 3}$