Simplify (x-3)/(x-1)+6/(x-3)

$\frac{x - 3}{x - 1} + \frac{6}{x - 3}$

To write $\frac{x - 3}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{x - 3}{x - 1} \cdot \frac{x - 3}{x - 3} + \frac{6}{x - 3}$

To write $\frac{6}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{x - 3}{x - 1} \cdot \frac{x - 3}{x - 3} + \frac{6}{x - 3} \cdot \frac{x - 1}{x - 1}$

Write each expression with a common denominator of $(x - 1) (x - 3)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x - 3}{x - 1}$ by $\frac{x - 3}{x - 3}$ .

$\frac{(x - 3) (x - 3)}{(x - 1) (x - 3)} + \frac{6}{x - 3} \cdot \frac{x - 1}{x - 1}$

Multiply $\frac{6}{x - 3}$ by $\frac{x - 1}{x - 1}$ .

$\frac{(x - 3) (x - 3)}{(x - 1) (x - 3)} + \frac{6 (x - 1)}{(x - 3) (x - 1)}$

Reorder the factors of $(x - 3) (x - 1)$ .

$\frac{(x - 3) (x - 3)}{(x - 1) (x - 3)} + \frac{6 (x - 1)}{(x - 1) (x - 3)}$

Combine the numerators over the common denominator.

$\frac{(x - 3) (x - 3) + 6 (x - 1)}{(x - 1) (x - 3)}$

Simplify the numerator.

Expand $(x - 3) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$\frac{x (x - 3) - 3 (x - 3) + 6 (x - 1)}{(x - 1) (x - 3)}$

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 3 - 3 (x - 3) + 6 (x - 1)}{(x - 1) (x - 3)}$

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 + 6 (x - 1)}{(x - 1) (x - 3)}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3 + 6 (x - 1)}{(x - 1) (x - 3)}$

Move $- 3$ to the left of $x$ .

$\frac{x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3 + 6 (x - 1)}{(x - 1) (x - 3)}$

Multiply $- 3$ by $- 3$ .

$\frac{x^{2} - 3 x - 3 x + 9 + 6 (x - 1)}{(x - 1) (x - 3)}$

Subtract $3 x$ from $- 3 x$ .

$\frac{x^{2} - 6 x + 9 + 6 (x - 1)}{(x - 1) (x - 3)}$

Apply the distributive property.

$\frac{x^{2} - 6 x + 9 + 6 x + 6 \cdot - 1}{(x - 1) (x - 3)}$

Multiply $6$ by $- 1$ .

$\frac{x^{2} - 6 x + 9 + 6 x - 6}{(x - 1) (x - 3)}$

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

$\frac{x^{2} + 0 + 9 - 6}{(x - 1) (x - 3)}$

Add $x^{2}$ and $0$ .

$\frac{x^{2} + 9 - 6}{(x - 1) (x - 3)}$

Subtract $6$ from $9$ .

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