Solve by Factoring x/(x^2-9)-1/(x-3)=1/(4x-12)

$\frac{x}{x^{2} - 9} - \frac{1}{x - 3} = \frac{1}{4 x - 12}$

Step 1

Subtract $\frac{1}{4 x - 12}$ from both sides of the equation.

$\frac{x}{x^{2} - 9} - \frac{1}{x - 3} - \frac{1}{4 x - 12} = 0$

Step 2

Simplify $\frac{x}{x^{2} - 9} - \frac{1}{x - 3} - \frac{1}{4 x - 12}$ .

Simplify each term.

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

$\frac{x}{x^{2} - 3^{2}} - \frac{1}{x - 3} - \frac{1}{4 x - 12} = 0$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$\frac{x}{(x + 3) (x - 3)} - \frac{1}{x - 3} - \frac{1}{4 x - 12} = 0$

Factor $4$ out of $4 x - 12$ .

Factor $4$ out of $4 x$ .

$\frac{x}{(x + 3) (x - 3)} - \frac{1}{x - 3} - \frac{1}{4 (x) - 12} = 0$

Factor $4$ out of $- 12$ .

$\frac{x}{(x + 3) (x - 3)} - \frac{1}{x - 3} - \frac{1}{4 x + 4 \cdot - 3} = 0$

Factor $4$ out of $4 x + 4 \cdot - 3$ .

$\frac{x}{(x + 3) (x - 3)} - \frac{1}{x - 3} - \frac{1}{4 (x - 3)} = 0$

Find the common denominator.

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

$\frac{x}{(x + 3) (x - 3)} \cdot \frac{4}{4} - \frac{1}{x - 3} - \frac{1}{4 (x - 3)} = 0$

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

$\frac{x \cdot 4}{(x + 3) (x - 3) \cdot 4} - \frac{1}{x - 3} - \frac{1}{4 (x - 3)} = 0$

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

$\frac{x \cdot 4}{(x + 3) (x - 3) \cdot 4} - (\frac{1}{x - 3} \cdot \frac{(x + 3) \cdot 4}{(x + 3) \cdot 4}) - \frac{1}{4 (x - 3)} = 0$

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

$\frac{x \cdot 4}{(x + 3) (x - 3) \cdot 4} - \frac{(x + 3) \cdot 4}{(x - 3) ((x + 3) \cdot 4)} - \frac{1}{4 (x - 3)} = 0$

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

$\frac{x \cdot 4}{(x + 3) (x - 3) \cdot 4} - \frac{(x + 3) \cdot 4}{(x - 3) ((x + 3) \cdot 4)} - (\frac{1}{4 (x - 3)} \cdot \frac{x + 3}{x + 3}) = 0$

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

$\frac{x \cdot 4}{(x + 3) (x - 3) \cdot 4} - \frac{(x + 3) \cdot 4}{(x - 3) ((x + 3) \cdot 4)} - \frac{x + 3}{4 (x - 3) (x + 3)} = 0$

Reorder the factors of $(x - 3) ((x + 3) \cdot 4)$ .

$\frac{x \cdot 4}{(x + 3) (x - 3) \cdot 4} - \frac{(x + 3) \cdot 4}{4 (x + 3) (x - 3)} - \frac{x + 3}{4 (x - 3) (x + 3)} = 0$

Reorder the factors of $4 (x - 3) (x + 3)$ .

$\frac{x \cdot 4}{(x + 3) (x - 3) \cdot 4} - \frac{(x + 3) \cdot 4}{4 (x + 3) (x - 3)} - \frac{x + 3}{4 (x + 3) (x - 3)} = 0$

Combine the numerators over the common denominator.

$\frac{x \cdot 4 - (x + 3) \cdot 4 - (x + 3)}{4 (x + 3) (x - 3)} = 0$

Simplify each term.

Move $4$ to the left of $x$ .

$\frac{4 \cdot x - (x + 3) \cdot 4 - (x + 3)}{4 (x + 3) (x - 3)} = 0$

Apply the distributive property.

$\frac{4 x + (- x - 1 \cdot 3) \cdot 4 - (x + 3)}{4 (x + 3) (x - 3)} = 0$

Multiply $- 1$ by $3$ .

$\frac{4 x + (- x - 3) \cdot 4 - (x + 3)}{4 (x + 3) (x - 3)} = 0$

Apply the distributive property.

$\frac{4 x - x \cdot 4 - 3 \cdot 4 - (x + 3)}{4 (x + 3) (x - 3)} = 0$

Multiply $4$ by $- 1$ .

$\frac{4 x - 4 x - 3 \cdot 4 - (x + 3)}{4 (x + 3) (x - 3)} = 0$

Multiply $- 3$ by $4$ .

$\frac{4 x - 4 x - 12 - (x + 3)}{4 (x + 3) (x - 3)} = 0$

Apply the distributive property.

$\frac{4 x - 4 x - 12 - x - 1 \cdot 3}{4 (x + 3) (x - 3)} = 0$

Multiply $- 1$ by $3$ .

$\frac{4 x - 4 x - 12 - x - 3}{4 (x + 3) (x - 3)} = 0$

Subtract $4 x$ from $4 x$ .

$\frac{0 - 12 - x - 3}{4 (x + 3) (x - 3)} = 0$

Subtract $12$ from $0$ .

$\frac{- 12 - x - 3}{4 (x + 3) (x - 3)} = 0$

Subtract $3$ from $- 12$ .

$\frac{- x - 15}{4 (x + 3) (x - 3)} = 0$

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

$\frac{- (x) - 15}{4 (x + 3) (x - 3)} = 0$

Rewrite $- 15$ as $- 1 (15)$ .

$\frac{- (x) - 1 (15)}{4 (x + 3) (x - 3)} = 0$

Factor $- 1$ out of $- (x) - 1 (15)$ .

$\frac{- (x + 15)}{4 (x + 3) (x - 3)} = 0$

Rewrite $- (x + 15)$ as $- 1 (x + 15)$ .

$\frac{- 1 (x + 15)}{4 (x + 3) (x - 3)} = 0$

Move the negative in front of the fraction.

$- \frac{x + 15}{4 (x + 3) (x - 3)} = 0$

Step 3

Set the numerator equal to zero.

$x + 15 = 0$

Step 4

Subtract $15$ from both sides of the equation.

$x = - 15$

Solve by Factoring x/(x^2-9)-1/(x-3)=1/(4x-12)

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