Precalculus Examples

$\frac{2}{x^{2} - x - 2} + \frac{10}{x^{2} + 2 x - 8}$

Step 1

Simplify each term.

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Step 1.1

Factor $x^{2} - x - 2$ 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 $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 1.1.2

Write the factored form using these integers.

$\frac{2}{(x - 2) (x + 1)} + \frac{10}{x^{2} + 2 x - 8}$

Step 1.2

Factor $x^{2} + 2 x - 8$ 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 $- 8$ and whose sum is $2$ .

$- 2, 4$

Step 1.2.2

Write the factored form using these integers.

$\frac{2}{(x - 2) (x + 1)} + \frac{10}{(x - 2) (x + 4)}$

Step 2

To write $\frac{2}{(x - 2) (x + 1)}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$\frac{2}{(x - 2) (x + 1)} \cdot \frac{x + 4}{x + 4} + \frac{10}{(x - 2) (x + 4)}$

Step 3

To write $\frac{10}{(x - 2) (x + 4)}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\frac{2}{(x - 2) (x + 1)} \cdot \frac{x + 4}{x + 4} + \frac{10}{(x - 2) (x + 4)} \cdot \frac{x + 1}{x + 1}$

Step 4

Write each expression with a common denominator of $(x - 2) (x + 1) (x + 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2}{(x - 2) (x + 1)}$ by $\frac{x + 4}{x + 4}$ .

$\frac{2 (x + 4)}{(x - 2) (x + 1) (x + 4)} + \frac{10}{(x - 2) (x + 4)} \cdot \frac{x + 1}{x + 1}$

Step 4.2

Multiply $\frac{10}{(x - 2) (x + 4)}$ by $\frac{x + 1}{x + 1}$ .

$\frac{2 (x + 4)}{(x - 2) (x + 1) (x + 4)} + \frac{10 (x + 1)}{(x - 2) (x + 4) (x + 1)}$

Step 4.3

Reorder the factors of $(x - 2) (x + 1) (x + 4)$ .

$\frac{2 (x + 4)}{(x + 1) (x + 4) (x - 2)} + \frac{10 (x + 1)}{(x - 2) (x + 4) (x + 1)}$

Step 4.4

Reorder the factors of $(x - 2) (x + 4) (x + 1)$ .

$\frac{2 (x + 4)}{(x + 1) (x + 4) (x - 2)} + \frac{10 (x + 1)}{(x + 1) (x + 4) (x - 2)}$

Step 5

Combine the numerators over the common denominator.

$\frac{2 (x + 4) + 10 (x + 1)}{(x + 1) (x + 4) (x - 2)}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $2$ out of $2 (x + 4) + 10 (x + 1)$ .

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Step 6.1.1

Factor $2$ out of $10 (x + 1)$ .

$\frac{2 (x + 4) + 2 (5 (x + 1))}{(x + 1) (x + 4) (x - 2)}$

Step 6.1.2

Factor $2$ out of $2 (x + 4) + 2 (5 (x + 1))$ .

$\frac{2 (x + 4 + 5 (x + 1))}{(x + 1) (x + 4) (x - 2)}$

Step 6.2

Apply the distributive property.

$\frac{2 (x + 4 + 5 x + 5 \cdot 1)}{(x + 1) (x + 4) (x - 2)}$

Step 6.3

Multiply $5$ by $1$ .

$\frac{2 (x + 4 + 5 x + 5)}{(x + 1) (x + 4) (x - 2)}$

Step 6.4

Add $x$ and $5 x$ .

$\frac{2 (6 x + 4 + 5)}{(x + 1) (x + 4) (x - 2)}$

Step 6.5

Add $4$ and $5$ .

$\frac{2 (6 x + 9)}{(x + 1) (x + 4) (x - 2)}$

Step 6.6

Factor $3$ out of $6 x + 9$ .

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Step 6.6.1

Factor $3$ out of $6 x$ .

$\frac{2 (3 (2 x) + 9)}{(x + 1) (x + 4) (x - 2)}$

Step 6.6.2

Factor $3$ out of $9$ .

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

Step 6.6.3

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

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

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

Step 6.7

Multiply $2$ by $3$ .

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