Algebra Examples

$\frac{4}{3 x + 6} + \frac{x + 1}{x^{2} - 4}$

Step 1

Simplify each term.

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Step 1.1

Factor $3$ out of $3 x + 6$ .

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Step 1.1.1

Factor $3$ out of $3 x$ .

$\frac{4}{3 (x) + 6} + \frac{x + 1}{x^{2} - 4}$

Step 1.1.2

Factor $3$ out of $6$ .

$\frac{4}{3 x + 3 \cdot 2} + \frac{x + 1}{x^{2} - 4}$

Step 1.1.3

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

$\frac{4}{3 (x + 2)} + \frac{x + 1}{x^{2} - 4}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Rewrite $4$ as $2^{2}$ .

$\frac{4}{3 (x + 2)} + \frac{x + 1}{x^{2} - 2^{2}}$

Step 1.2.2

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 $b = 2$ .

$\frac{4}{3 (x + 2)} + \frac{x + 1}{(x + 2) (x - 2)}$

Step 2

To write $\frac{4}{3 (x + 2)}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{4}{3 (x + 2)} \cdot \frac{x - 2}{x - 2} + \frac{x + 1}{(x + 2) (x - 2)}$

Step 3

To write $\frac{x + 1}{(x + 2) (x - 2)}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{4}{3 (x + 2)} \cdot \frac{x - 2}{x - 2} + \frac{x + 1}{(x + 2) (x - 2)} \cdot \frac{3}{3}$

Step 4

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

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Step 4.1

Multiply $\frac{4}{3 (x + 2)}$ by $\frac{x - 2}{x - 2}$ .

$\frac{4 (x - 2)}{3 (x + 2) (x - 2)} + \frac{x + 1}{(x + 2) (x - 2)} \cdot \frac{3}{3}$

Step 4.2

Multiply $\frac{x + 1}{(x + 2) (x - 2)}$ by $\frac{3}{3}$ .

$\frac{4 (x - 2)}{3 (x + 2) (x - 2)} + \frac{(x + 1) \cdot 3}{(x + 2) (x - 2) \cdot 3}$

Step 4.3

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

$\frac{4 (x - 2)}{3 (x + 2) (x - 2)} + \frac{(x + 1) \cdot 3}{3 (x + 2) (x - 2)}$

Step 5

Combine the numerators over the common denominator.

$\frac{4 (x - 2) + (x + 1) \cdot 3}{3 (x + 2) (x - 2)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{4 x + 4 \cdot - 2 + (x + 1) \cdot 3}{3 (x + 2) (x - 2)}$

Step 6.2

Multiply $4$ by $- 2$ .

$\frac{4 x - 8 + (x + 1) \cdot 3}{3 (x + 2) (x - 2)}$

Step 6.3

Apply the distributive property.

$\frac{4 x - 8 + x \cdot 3 + 1 \cdot 3}{3 (x + 2) (x - 2)}$

Step 6.4

Move $3$ to the left of $x$ .

$\frac{4 x - 8 + 3 \cdot x + 1 \cdot 3}{3 (x + 2) (x - 2)}$

Step 6.5

Multiply $3$ by $1$ .

$\frac{4 x - 8 + 3 \cdot x + 3}{3 (x + 2) (x - 2)}$

Step 6.6

Add $4 x$ and $3 x$ .

$\frac{7 x - 8 + 3}{3 (x + 2) (x - 2)}$

Step 6.7

Add $- 8$ and $3$ .

$\frac{7 x - 5}{3 (x + 2) (x - 2)}$