Simplify 3/(x-2)+(x-2)/(x^2-4x+4)

$\frac{3}{x - 2} + \frac{x - 2}{x^{2} - 4 x + 4}$

Step 1

Simplify each term.

Factor using the perfect square rule.

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

$\frac{3}{x - 2} + \frac{x - 2}{x^{2} - 4 x + 2^{2}}$

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

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

Rewrite the polynomial.

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

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

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

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

Multiply by $1$ .

$\frac{3}{x - 2} + \frac{(x - 2) \cdot 1}{{(x - 2)}^{2}}$

Cancel the common factors.

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

$\frac{3}{x - 2} + \frac{(x - 2) \cdot 1}{(x - 2) (x - 2)}$

Cancel the common factor.

$\frac{3}{x - 2} + \frac{(x - 2) \cdot 1}{(x - 2) (x - 2)}$

Rewrite the expression.

$\frac{3}{x - 2} + \frac{1}{x - 2}$

Step 2

Combine fractions.

Combine the numerators over the common denominator.

$\frac{3 + 1}{x - 2}$

Add $3$ and $1$ .

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