Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

Step 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 $b = 3$ .

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

Step 2

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

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

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

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

Step 3.2

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Factor $x$ out of $6 x - x (x + 3)$ .

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Step 5.1.1

Factor $x$ out of $6 x$ .

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

Step 5.1.2

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

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

Step 5.1.3

Factor $x$ out of $x \cdot 6 + x (- (x + 3))$ .

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Multiply $- 1$ by $3$ .

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

Step 5.4

Subtract $3$ from $6$ .

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

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $- x + 3$ and $x - 3$ .

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Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.1.3

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

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

Step 6.1.4

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

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

Step 6.1.5

Cancel the common factor.

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

Step 6.1.6

Rewrite the expression.

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

Step 6.2

Simplify the expression.

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Step 6.2.1

Move $- 1$ to the left of $x$ .

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

Step 6.2.2

Move the negative in front of the fraction.

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