Algebra Examples

$\frac{2}{x - 2} + \frac{x}{x + 9} - \frac{x + 20}{x^{2} + 7 x - 18}$

Step 1

Factor $x^{2} + 7 x - 18$ 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 $- 18$ and whose sum is $7$ .

$- 2, 9$

Step 1.2

Write the factored form using these integers.

$\frac{2}{x - 2} + \frac{x}{x + 9} - \frac{x + 20}{(x - 2) (x + 9)}$

Step 2

To write $\frac{2}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x + 9}{x + 9}$ .

$\frac{2}{x - 2} \cdot \frac{x + 9}{x + 9} + \frac{x}{x + 9} - \frac{x + 20}{(x - 2) (x + 9)}$

Step 3

To write $\frac{x}{x + 9}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{2}{x - 2} \cdot \frac{x + 9}{x + 9} + \frac{x}{x + 9} \cdot \frac{x - 2}{x - 2} - \frac{x + 20}{(x - 2) (x + 9)}$

Step 4

Write each expression with a common denominator of $(x - 2) (x + 9)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2}{x - 2}$ by $\frac{x + 9}{x + 9}$ .

$\frac{2 (x + 9)}{(x - 2) (x + 9)} + \frac{x}{x + 9} \cdot \frac{x - 2}{x - 2} - \frac{x + 20}{(x - 2) (x + 9)}$

Step 4.2

Multiply $\frac{x}{x + 9}$ by $\frac{x - 2}{x - 2}$ .

$\frac{2 (x + 9)}{(x - 2) (x + 9)} + \frac{x (x - 2)}{(x + 9) (x - 2)} - \frac{x + 20}{(x - 2) (x + 9)}$

Step 4.3

Reorder the factors of $(x - 2) (x + 9)$ .

$\frac{2 (x + 9)}{(x + 9) (x - 2)} + \frac{x (x - 2)}{(x + 9) (x - 2)} - \frac{x + 20}{(x - 2) (x + 9)}$

Step 5

Simplify terms.

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Step 5.1

Combine the numerators over the common denominator.

$\frac{2 (x + 9) + x (x - 2)}{(x + 9) (x - 2)} - \frac{x + 20}{(x - 2) (x + 9)}$

Step 5.2

Combine the numerators over the common denominator.

$\frac{2 (x + 9) + x (x - 2) - (x + 20)}{(x + 9) (x - 2)}$

Step 5.3

Simplify each term.

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Step 5.3.1

Apply the distributive property.

$\frac{2 x + 2 \cdot 9 + x (x - 2) - (x + 20)}{(x + 9) (x - 2)}$

Step 5.3.2

Multiply $2$ by $9$ .

$\frac{2 x + 18 + x (x - 2) - (x + 20)}{(x + 9) (x - 2)}$

Step 5.3.3

Apply the distributive property.

$\frac{2 x + 18 + x \cdot x + x \cdot - 2 - (x + 20)}{(x + 9) (x - 2)}$

Step 5.3.4

Multiply $x$ by $x$ .

$\frac{2 x + 18 + x^{2} + x \cdot - 2 - (x + 20)}{(x + 9) (x - 2)}$

Step 5.3.5

Move $- 2$ to the left of $x$ .

$\frac{2 x + 18 + x^{2} - 2 \cdot x - (x + 20)}{(x + 9) (x - 2)}$

Step 5.3.6

Apply the distributive property.

$\frac{2 x + 18 + x^{2} - 2 x - x - 1 \cdot 20}{(x + 9) (x - 2)}$

Step 5.3.7

Multiply $- 1$ by $20$ .

$\frac{2 x + 18 + x^{2} - 2 x - x - 20}{(x + 9) (x - 2)}$

Step 5.4

Simplify by adding terms.

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Step 5.4.1

Combine the opposite terms in $2 x + 18 + x^{2} - 2 x - x - 20$ .

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Step 5.4.1.1

Subtract $2 x$ from $2 x$ .

$\frac{18 + x^{2} + 0 - x - 20}{(x + 9) (x - 2)}$

Step 5.4.1.2

Add $18 + x^{2}$ and $0$ .

$\frac{18 + x^{2} - x - 20}{(x + 9) (x - 2)}$

Step 5.4.2

Subtract $20$ from $18$ .

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

Step 6

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

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Step 6.1

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 6.2

Write the factored form using these integers.

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

Step 7

Cancel the common factor of $x - 2$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

$\frac{x + 1}{x + 9}$