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

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

Step 1

Factor using the perfect square rule.

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

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

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

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

Rewrite the polynomial.

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

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

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

Step 2

Simplify the denominator.

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

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

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 $b = 2$ .

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

Step 3

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

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

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

Cancel the common factors.

Factor $x - 2$ out of $(x + 2) (x - 2)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{x - 2}{x + 2}$