Simplify (x^2-6x)/(x^2-4x+4)+(4x)/(x^2-4x+4)

$\frac{x^{2} - 6 x}{x^{2} - 4 x + 4} + \frac{4 x}{x^{2} - 4 x + 4}$

Step 1

Simplify terms.

Combine the numerators over the common denominator.

$\frac{x^{2} - 6 x + 4 x}{x^{2} - 4 x + 4}$

Add $- 6 x$ and $4 x$ .

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

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

Factor $x$ out of $x^{2}$ .

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

Factor $x$ out of $- 2 x$ .

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

Factor $x$ out of $x \cdot x + x \cdot - 2$ .

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

Step 2

Factor using the perfect square rule.

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

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

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.

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

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

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

Step 3

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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