Algebra Examples

$\frac{x + 4}{x^{2} - x - 12} + \frac{2 x}{x - 4}$

Step 1

Factor $x^{2} - x - 12$ using the AC method.

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Step 1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $- 1$ .

$- 4, 3$

Step 1.2

Write the factored form using these integers.

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $x$ by $x$ by adding the exponents.

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Step 4.2.1

Move $x$ .

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

Step 4.2.2

Multiply $x$ by $x$ .

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

Step 4.3

Multiply $3$ by $2$ .

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

Step 4.4

Add $x$ and $6 x$ .

$\frac{7 x + 4 + 2 x^{2}}{(x - 4) (x + 3)}$

Step 4.5

Reorder terms.

$\frac{2 x^{2} + 7 x + 4}{(x - 4) (x + 3)}$