Algebra Examples

$\frac{- 3 x}{x^{2} - 9} + \frac{4}{2 x - 6}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

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

$\frac{- 3 x}{x^{2} - 3^{2}} + \frac{4}{2 x - 6}$

Step 1.1.2

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 $b = 3$ .

$\frac{- 3 x}{(x + 3) (x - 3)} + \frac{4}{2 x - 6}$

Step 1.2

Move the negative in front of the fraction.

$- \frac{3 x}{(x + 3) (x - 3)} + \frac{4}{2 x - 6}$

Step 1.3

Cancel the common factor of $4$ and $2 x - 6$ .

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Step 1.3.1

Factor $2$ out of $4$ .

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

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor $2$ out of $2 x$ .

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

Step 1.3.2.2

Factor $2$ out of $- 6$ .

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

Step 1.3.2.3

Factor $2$ out of $2 (x) + 2 (- 3)$ .

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

Step 1.3.2.4

Cancel the common factor.

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

Step 1.3.2.5

Rewrite the expression.

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

Step 2

To write $\frac{2}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

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

Step 3

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

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Step 3.1

Multiply $\frac{2}{x - 3}$ by $\frac{x + 3}{x + 3}$ .

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

Step 3.2

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $2$ by $3$ .

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

Step 5.3

Add $- 3 x$ and $2 x$ .

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

Step 6

Simplify with factoring out.

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Step 6.1

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

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

Step 6.2

Rewrite $6$ as $- 1 (- 6)$ .

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

Step 6.3

Factor $- 1$ out of $- (x) - 1 (- 6)$ .

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

Step 6.4

Simplify the expression.

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Step 6.4.1

Rewrite $- (x - 6)$ as $- 1 (x - 6)$ .

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

Step 6.4.2

Move the negative in front of the fraction.

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