Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Factor $2$ out of $2 x + 6$ .

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Step 1.1.1

Factor $2$ out of $2 x$ .

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

Step 1.1.2

Factor $2$ out of $6$ .

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

Step 1.1.3

Factor $2$ out of $2 x + 2 \cdot 3$ .

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

Step 1.2

Factor $2 x$ out of $2 x^{2} + 6 x$ .

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Step 1.2.1

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

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

Step 1.2.2

Factor $2 x$ out of $6 x$ .

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

Step 1.2.3

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

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

Step 2

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

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

Step 3

Write each expression with a common denominator of $2 (x + 3) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

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

Step 3.2

Reorder the factors of $2 (x + 3) x$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $- 1$ by $- 6$ .

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

Step 5.3

Subtract $x$ from $3 x$ .

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

Step 5.4

Factor $2$ out of $2 x + 6$ .

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Step 5.4.1

Factor $2$ out of $2 x$ .

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

Step 5.4.2

Factor $2$ out of $6$ .

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

Step 5.4.3

Factor $2$ out of $2 x + 2 \cdot 3$ .

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

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $2$ .

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Step 6.1.1

Cancel the common factor.

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

Step 6.1.2

Rewrite the expression.

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

Step 6.2

Cancel the common factor of $x + 3$ .

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

$\frac{1}{x}$