Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

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

$- 3, 1$

Step 1.1.2

Write the factored form using these integers.

$\frac{- 2}{(x - 3) (x + 1)} + \frac{3}{x^{2} - 9}$

Step 1.2

Move the negative in front of the fraction.

$- \frac{2}{(x - 3) (x + 1)} + \frac{3}{x^{2} - 9}$

Step 1.3

Simplify the denominator.

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Step 1.3.1

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

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

Step 1.3.2

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 $b = 3$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $(x - 3) (x + 1) (x + 3)$ .

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

Step 4.4

Reorder the factors of $(x + 3) (x - 3) (x + 1)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $- 2$ by $3$ .

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

Step 6.3

Apply the distributive property.

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

Step 6.4

Multiply $3$ by $1$ .

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

Step 6.5

Add $- 2 x$ and $3 x$ .

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

Step 6.6

Add $- 6$ and $3$ .

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

Step 7

Cancel the common factor of $x - 3$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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