Algebra Examples

$\frac{3}{x + 1} + \frac{4}{2 x - 6} - \frac{x^{2} - 5}{x^{2} - 2 x - 3}$

Step 1

Simplify each term.

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Step 1.1

Cancel the common factor of $4$ and $2 x - 6$ .

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Step 1.1.1

Factor $2$ out of $4$ .

$\frac{3}{x + 1} + \frac{2 \cdot 2}{2 x - 6} - \frac{x^{2} - 5}{x^{2} - 2 x - 3}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $2$ out of $2 x$ .

$\frac{3}{x + 1} + \frac{2 \cdot 2}{2 (x) - 6} - \frac{x^{2} - 5}{x^{2} - 2 x - 3}$

Step 1.1.2.2

Factor $2$ out of $- 6$ .

$\frac{3}{x + 1} + \frac{2 \cdot 2}{2 (x) + 2 (- 3)} - \frac{x^{2} - 5}{x^{2} - 2 x - 3}$

Step 1.1.2.3

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

$\frac{3}{x + 1} + \frac{2 \cdot 2}{2 (x - 3)} - \frac{x^{2} - 5}{x^{2} - 2 x - 3}$

Step 1.1.2.4

Cancel the common factor.

$\frac{3}{x + 1} + \frac{2 \cdot 2}{2 (x - 3)} - \frac{x^{2} - 5}{x^{2} - 2 x - 3}$

Step 1.1.2.5

Rewrite the expression.

$\frac{3}{x + 1} + \frac{2}{x - 3} - \frac{x^{2} - 5}{x^{2} - 2 x - 3}$

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.

$\frac{3}{x + 1} + \frac{2}{x - 3} - \frac{x^{2} - 5}{(x - 3) (x + 1)}$

Step 2

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

$\frac{3}{x + 1} \cdot \frac{x - 3}{x - 3} + \frac{2}{x - 3} - \frac{x^{2} - 5}{(x - 3) (x + 1)}$

Step 3

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

$\frac{3}{x + 1} \cdot \frac{x - 3}{x - 3} + \frac{2}{x - 3} \cdot \frac{x + 1}{x + 1} - \frac{x^{2} - 5}{(x - 3) (x + 1)}$

Step 4

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

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Step 4.1

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

$\frac{3 (x - 3)}{(x + 1) (x - 3)} + \frac{2}{x - 3} \cdot \frac{x + 1}{x + 1} - \frac{x^{2} - 5}{(x - 3) (x + 1)}$

Step 4.2

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

$\frac{3 (x - 3)}{(x + 1) (x - 3)} + \frac{2 (x + 1)}{(x - 3) (x + 1)} - \frac{x^{2} - 5}{(x - 3) (x + 1)}$

Step 4.3

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

$\frac{3 (x - 3)}{(x + 1) (x - 3)} + \frac{2 (x + 1)}{(x + 1) (x - 3)} - \frac{x^{2} - 5}{(x - 3) (x + 1)}$

Step 5

Simplify terms.

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Step 5.1

Combine the numerators over the common denominator.

$\frac{3 (x - 3) + 2 (x + 1)}{(x + 1) (x - 3)} - \frac{x^{2} - 5}{(x - 3) (x + 1)}$

Step 5.2

Combine the numerators over the common denominator.

$\frac{3 (x - 3) + 2 (x + 1) - (x^{2} - 5)}{(x + 1) (x - 3)}$

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

$\frac{3 x + 3 \cdot - 3 + 2 (x + 1) - (x^{2} - 5)}{(x + 1) (x - 3)}$

Step 6.2

Multiply $3$ by $- 3$ .

$\frac{3 x - 9 + 2 (x + 1) - (x^{2} - 5)}{(x + 1) (x - 3)}$

Step 6.3

Apply the distributive property.

$\frac{3 x - 9 + 2 x + 2 \cdot 1 - (x^{2} - 5)}{(x + 1) (x - 3)}$

Step 6.4

Multiply $2$ by $1$ .

$\frac{3 x - 9 + 2 x + 2 - (x^{2} - 5)}{(x + 1) (x - 3)}$

Step 6.5

Apply the distributive property.

$\frac{3 x - 9 + 2 x + 2 - x^{2} - - 5}{(x + 1) (x - 3)}$

Step 6.6

Multiply $- 1$ by $- 5$ .

$\frac{3 x - 9 + 2 x + 2 - x^{2} + 5}{(x + 1) (x - 3)}$

Step 7

Simplify terms.

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Step 7.1

Add $3 x$ and $2 x$ .

$\frac{5 x - 9 + 2 - x^{2} + 5}{(x + 1) (x - 3)}$

Step 7.2

Simplify the expression.

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Step 7.2.1

Add $- 9$ and $2$ .

$\frac{5 x - 7 - x^{2} + 5}{(x + 1) (x - 3)}$

Step 7.2.2

Add $- 7$ and $5$ .

$\frac{5 x - x^{2} - 2}{(x + 1) (x - 3)}$

Step 7.2.3

Reorder $5 x$ and $- x^{2}$ .

$\frac{- x^{2} + 5 x - 2}{(x + 1) (x - 3)}$

Step 7.3

Factor $- 1$ out of $- x^{2}$ .

$\frac{- (x^{2}) + 5 x - 2}{(x + 1) (x - 3)}$

Step 7.4

Factor $- 1$ out of $5 x$ .

$\frac{- (x^{2}) - (- 5 x) - 2}{(x + 1) (x - 3)}$

Step 7.5

Factor $- 1$ out of $- (x^{2}) - (- 5 x)$ .

$\frac{- (x^{2} - 5 x) - 2}{(x + 1) (x - 3)}$

Step 7.6

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{- (x^{2} - 5 x) - 1 (2)}{(x + 1) (x - 3)}$

Step 7.7

Factor $- 1$ out of $- (x^{2} - 5 x) - 1 (2)$ .

$\frac{- (x^{2} - 5 x + 2)}{(x + 1) (x - 3)}$

Step 7.8

Simplify the expression.

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Step 7.8.1

Rewrite $- (x^{2} - 5 x + 2)$ as $- 1 (x^{2} - 5 x + 2)$ .

$\frac{- 1 (x^{2} - 5 x + 2)}{(x + 1) (x - 3)}$

Step 7.8.2

Move the negative in front of the fraction.

$- \frac{x^{2} - 5 x + 2}{(x + 1) (x - 3)}$