Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Factor $2$ out of $2 x$ .

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

Step 1.1.2

Factor $2$ out of $2$ .

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

Step 1.1.3

Factor $2$ out of $2 (x) + 2 (1)$ .

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

Step 1.2

Factor $x^{2} - 2 x - 3$ using the AC method.

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Step 1.2.1

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 1.2.2

Write the factored form using these integers.

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

Step 1.3

Cancel the common factor of $x + 1$ .

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Rewrite the expression.

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

Step 2

Combine the numerators over the common denominator.

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