Pre-Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

$2, 3$

Step 1.1.2

Write the factored form using these integers.

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

Step 1.2

Simplify the denominator.

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Step 1.2.1

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

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

Step 1.2.2

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 $b = 2$ .

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

Step 2

To write $\frac{x - 4}{(x + 2) (x + 3)}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

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

Step 3

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

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

Step 4

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

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Step 4.1

Multiply $\frac{x - 4}{(x + 2) (x + 3)}$ by $\frac{x - 2}{x - 2}$ .

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Expand $(x - 4) (x - 2)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Apply the distributive property.

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

Step 6.1.3

Apply the distributive property.

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

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 $x$ .

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

Step 6.2.1.2

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

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

Step 6.2.1.3

Multiply $- 4$ by $- 2$ .

$\frac{x^{2} - 2 x - 4 x + 8 + (x - 1) (x + 3)}{(x + 2) (x + 3) (x - 2)}$

Step 6.2.2

Subtract $4 x$ from $- 2 x$ .

$\frac{x^{2} - 6 x + 8 + (x - 1) (x + 3)}{(x + 2) (x + 3) (x - 2)}$

Step 6.3

Expand $(x - 1) (x + 3)$ using the FOIL Method.

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Step 6.3.1

Apply the distributive property.

$\frac{x^{2} - 6 x + 8 + x (x + 3) - 1 (x + 3)}{(x + 2) (x + 3) (x - 2)}$

Step 6.3.2

Apply the distributive property.

$\frac{x^{2} - 6 x + 8 + x \cdot x + x \cdot 3 - 1 (x + 3)}{(x + 2) (x + 3) (x - 2)}$

Step 6.3.3

Apply the distributive property.

$\frac{x^{2} - 6 x + 8 + x \cdot x + x \cdot 3 - 1 x - 1 \cdot 3}{(x + 2) (x + 3) (x - 2)}$

Step 6.4

Simplify and combine like terms.

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Step 6.4.1

Simplify each term.

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Step 6.4.1.1

Multiply $x$ by $x$ .

$\frac{x^{2} - 6 x + 8 + x^{2} + x \cdot 3 - 1 x - 1 \cdot 3}{(x + 2) (x + 3) (x - 2)}$

Step 6.4.1.2

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

$\frac{x^{2} - 6 x + 8 + x^{2} + 3 \cdot x - 1 x - 1 \cdot 3}{(x + 2) (x + 3) (x - 2)}$

Step 6.4.1.3

Rewrite $- 1 x$ as $- x$ .

$\frac{x^{2} - 6 x + 8 + x^{2} + 3 x - x - 1 \cdot 3}{(x + 2) (x + 3) (x - 2)}$

Step 6.4.1.4

Multiply $- 1$ by $3$ .

$\frac{x^{2} - 6 x + 8 + x^{2} + 3 x - x - 3}{(x + 2) (x + 3) (x - 2)}$

Step 6.4.2

Subtract $x$ from $3 x$ .

$\frac{x^{2} - 6 x + 8 + x^{2} + 2 x - 3}{(x + 2) (x + 3) (x - 2)}$

Step 6.5

Add $x^{2}$ and $x^{2}$ .

$\frac{2 x^{2} - 6 x + 8 + 2 x - 3}{(x + 2) (x + 3) (x - 2)}$

Step 6.6

Add $- 6 x$ and $2 x$ .

$\frac{2 x^{2} - 4 x + 8 - 3}{(x + 2) (x + 3) (x - 2)}$

Step 6.7

Subtract $3$ from $8$ .

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