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

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

Step 1

Simplify the numerator.

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

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

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

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

Step 2

Factor using the perfect square rule.

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

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

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

$4 p = 2 \cdot p \cdot 2$

Rewrite the polynomial.

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

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

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

Step 3

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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