Pre-Algebra Examples

$\frac{x + 9}{x - 2} + \frac{- 8 x - 39}{x^{2} + x - 6}$

Step 1

Factor $x^{2} + x - 6$ 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 $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 1.2

Write the factored form using these integers.

$\frac{x + 9}{x - 2} + \frac{- 8 x - 39}{(x - 2) (x + 3)}$

Step 2

To write $\frac{x + 9}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{x + 9}{x - 2} \cdot \frac{x + 3}{x + 3} + \frac{- 8 x - 39}{(x - 2) (x + 3)}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{x + 9}{x - 2}$ by $\frac{x + 3}{x + 3}$ .

$\frac{(x + 9) (x + 3)}{(x - 2) (x + 3)} + \frac{- 8 x - 39}{(x - 2) (x + 3)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{(x + 9) (x + 3) - 8 x - 39}{(x - 2) (x + 3)}$

Step 4

Simplify the numerator.

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Step 4.1

Expand $(x + 9) (x + 3)$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

$\frac{x (x + 3) + 9 (x + 3) - 8 x - 39}{(x - 2) (x + 3)}$

Step 4.1.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot 3 + 9 (x + 3) - 8 x - 39}{(x - 2) (x + 3)}$

Step 4.1.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot 3 + 9 x + 9 \cdot 3 - 8 x - 39}{(x - 2) (x + 3)}$

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 3 + 9 x + 9 \cdot 3 - 8 x - 39}{(x - 2) (x + 3)}$

Step 4.2.1.2

Move $3$ to the left of $x$ .

$\frac{x^{2} + 3 \cdot x + 9 x + 9 \cdot 3 - 8 x - 39}{(x - 2) (x + 3)}$

Step 4.2.1.3

Multiply $9$ by $3$ .

$\frac{x^{2} + 3 x + 9 x + 27 - 8 x - 39}{(x - 2) (x + 3)}$

Step 4.2.2

Add $3 x$ and $9 x$ .

$\frac{x^{2} + 12 x + 27 - 8 x - 39}{(x - 2) (x + 3)}$

Step 4.3

Subtract $8 x$ from $12 x$ .

$\frac{x^{2} + 4 x + 27 - 39}{(x - 2) (x + 3)}$

Step 4.4

Subtract $39$ from $27$ .

$\frac{x^{2} + 4 x - 12}{(x - 2) (x + 3)}$

Step 4.5

Factor $x^{2} + 4 x - 12$ using the AC method.

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Step 4.5.1

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 $- 12$ and whose sum is $4$ .

$- 2, 6$

Step 4.5.2

Write the factored form using these integers.

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

Step 5

Cancel the common factor of $x - 2$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

$\frac{x + 6}{x + 3}$