Algebra Examples

$\frac{x + 2}{x + 3} - \frac{5}{x^{2} + x - 6} + \frac{1}{2 - x}$

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.

$\frac{x + 2}{x + 3} - \frac{5}{(x - 2) (x + 3)} + \frac{1}{2 - x}$

Step 2

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

$\frac{x + 2}{x + 3} \cdot \frac{x - 2}{x - 2} - \frac{5}{(x - 2) (x + 3)} + \frac{1}{2 - x}$

Step 3

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

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Step 3.1

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

$\frac{(x + 2) (x - 2)}{(x + 3) (x - 2)} - \frac{5}{(x - 2) (x + 3)} + \frac{1}{2 - x}$

Step 3.2

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

$\frac{(x + 2) (x - 2)}{(x + 3) (x - 2)} - \frac{5}{(x + 3) (x - 2)} + \frac{1}{2 - x}$

Step 4

Simplify terms.

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Step 4.1

Combine the numerators over the common denominator.

$\frac{(x + 2) (x - 2) - 5}{(x + 3) (x - 2)} + \frac{1}{2 - x}$

Step 4.2

Simplify each term.

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Step 4.2.1

Simplify the numerator.

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Step 4.2.1.1

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

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Step 4.2.1.1.1

Apply the distributive property.

$\frac{x (x - 2) + 2 (x - 2) - 5}{(x + 3) (x - 2)} + \frac{1}{2 - x}$

Step 4.2.1.1.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 2 + 2 (x - 2) - 5}{(x + 3) (x - 2)} + \frac{1}{2 - x}$

Step 4.2.1.1.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2 - 5}{(x + 3) (x - 2)} + \frac{1}{2 - x}$

Step 4.2.1.2

Combine the opposite terms in $x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2$ .

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Step 4.2.1.2.1

Reorder the factors in the terms $x \cdot - 2$ and $2 x$ .

$\frac{x \cdot x - 2 x + 2 x + 2 \cdot - 2 - 5}{(x + 3) (x - 2)} + \frac{1}{2 - x}$

Step 4.2.1.2.2

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

$\frac{x \cdot x + 0 + 2 \cdot - 2 - 5}{(x + 3) (x - 2)} + \frac{1}{2 - x}$

Step 4.2.1.2.3

Add $x \cdot x$ and $0$ .

$\frac{x \cdot x + 2 \cdot - 2 - 5}{(x + 3) (x - 2)} + \frac{1}{2 - x}$

Step 4.2.1.3

Simplify each term.

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Step 4.2.1.3.1

Multiply $x$ by $x$ .

$\frac{x^{2} + 2 \cdot - 2 - 5}{(x + 3) (x - 2)} + \frac{1}{2 - x}$

Step 4.2.1.3.2

Multiply $2$ by $- 2$ .

$\frac{x^{2} - 4 - 5}{(x + 3) (x - 2)} + \frac{1}{2 - x}$

Step 4.2.1.4

Subtract $5$ from $- 4$ .

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

Step 4.2.1.5

Rewrite $x^{2} - 9$ in a factored form.

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Step 4.2.1.5.1

Rewrite $9$ as $3^{2}$ .

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

Step 4.2.1.5.2

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 $b = 3$ .

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

Step 4.2.2

Cancel the common factor of $x + 3$ .

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Step 4.2.2.1

Cancel the common factor.

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

Step 4.2.2.2

Rewrite the expression.

$\frac{x - 3}{x - 2} + \frac{1}{2 - x}$

Step 4.3

Simplify terms.

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Step 4.3.1

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 4.3.2

Factor $- 1$ out of $- x$ .

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

Step 4.3.3

Factor $- 1$ out of $- 1 (- 2) - (x)$ .

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

Step 4.3.4

Simplify the expression.

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Step 4.3.4.1

Move a negative from the denominator of $\frac{1}{- 1 (- 2 + x)}$ to the numerator.

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

Step 4.3.4.2

Reorder terms.

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

Step 4.3.5

Combine the numerators over the common denominator.

$\frac{x - 3 - 1 \cdot 1}{x - 2}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 1$ by $1$ .

$\frac{x - 3 - 1}{x - 2}$

Step 5.2

Subtract $1$ from $- 3$ .

$\frac{x - 4}{x - 2}$