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

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

Step 1

Simplify each term.

Factor $x$ out of $x^{2} - 3 x$ .

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

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

Factor $x$ out of $- 3 x$ .

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

Factor $x$ out of $x \cdot x + x \cdot - 3$ .

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

Factor $x^{2} - 4 x + 3$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $- 4$ .

$- 3, - 1$

Write the factored form using these integers.

$\frac{3}{x (x - 3)} - \frac{4}{(x - 3) (x - 1)}$

Step 2

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

$\frac{3}{x (x - 3)} \cdot \frac{x - 1}{x - 1} - \frac{4}{(x - 3) (x - 1)}$

Step 3

To write $- \frac{4}{(x - 3) (x - 1)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{3}{x (x - 3)} \cdot \frac{x - 1}{x - 1} - \frac{4}{(x - 3) (x - 1)} \cdot \frac{x}{x}$

Step 4

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

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

$\frac{3 (x - 1)}{x (x - 3) (x - 1)} - \frac{4}{(x - 3) (x - 1)} \cdot \frac{x}{x}$

Multiply $\frac{4}{(x - 3) (x - 1)}$ by $\frac{x}{x}$ .

$\frac{3 (x - 1)}{x (x - 3) (x - 1)} - \frac{4 x}{(x - 3) (x - 1) x}$

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

$\frac{3 (x - 1)}{x (x - 1) (x - 3)} - \frac{4 x}{(x - 3) (x - 1) x}$

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

$\frac{3 (x - 1)}{x (x - 1) (x - 3)} - \frac{4 x}{x (x - 1) (x - 3)}$

Step 5

Combine the numerators over the common denominator.

$\frac{3 (x - 1) - 4 x}{x (x - 1) (x - 3)}$

Step 6

Simplify the numerator.

Apply the distributive property.

$\frac{3 x + 3 \cdot - 1 - 4 x}{x (x - 1) (x - 3)}$

Multiply $3$ by $- 1$ .

$\frac{3 x - 3 - 4 x}{x (x - 1) (x - 3)}$

Subtract $4 x$ from $3 x$ .

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

Step 7

Simplify with factoring out.

Factor $- 1$ out of $- x$ .

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

Rewrite $- 3$ as $- 1 (3)$ .

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

Factor $- 1$ out of $- (x) - 1 (3)$ .

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

Simplify the expression.

Rewrite $- (x + 3)$ as $- 1 (x + 3)$ .

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

Move the negative in front of the fraction.

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

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

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