Pre-Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Rewrite $4$ as $2^{2}$ .

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

Step 1.1.2

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 $b = 2$ .

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

Step 1.2

Factor $x^{2} - 5 x + 6$ 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 $6$ and whose sum is $- 5$ .

$- 3, - 2$

Step 1.2.2

Write the factored form using these integers.

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

Step 2

To write $\frac{x + 6}{(x + 2) (x - 2)}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

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

Step 3

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

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

Step 4

Write each expression with a common denominator of $(x + 2) (x - 2) (x - 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $(x - 3) (x - 2) (x + 2)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Expand $(x + 6) (x - 3)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Apply the distributive property.

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

Step 6.1.3

Apply the distributive property.

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

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $x$ by $x$ .

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

Step 6.2.1.2

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

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

Step 6.2.1.3

Multiply $6$ by $- 3$ .

$\frac{x^{2} - 3 x + 6 x - 18 + 2 (x + 2)}{(x + 2) (x - 2) (x - 3)}$

Step 6.2.2

Add $- 3 x$ and $6 x$ .

$\frac{x^{2} + 3 x - 18 + 2 (x + 2)}{(x + 2) (x - 2) (x - 3)}$

Step 6.3

Apply the distributive property.

$\frac{x^{2} + 3 x - 18 + 2 x + 2 \cdot 2}{(x + 2) (x - 2) (x - 3)}$

Step 6.4

Multiply $2$ by $2$ .

$\frac{x^{2} + 3 x - 18 + 2 x + 4}{(x + 2) (x - 2) (x - 3)}$

Step 6.5

Add $3 x$ and $2 x$ .

$\frac{x^{2} + 5 x - 18 + 4}{(x + 2) (x - 2) (x - 3)}$

Step 6.6

Add $- 18$ and $4$ .

$\frac{x^{2} + 5 x - 14}{(x + 2) (x - 2) (x - 3)}$

Step 6.7

Factor $x^{2} + 5 x - 14$ using the AC method.

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Step 6.7.1

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 $- 14$ and whose sum is $5$ .

$- 2, 7$

Step 6.7.2

Write the factored form using these integers.

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

Step 7

Cancel the common factor of $x - 2$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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