Algebra Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $(x - 2) (x^{2} - 4) (x + 2)$ .

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x - 2$ .

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Step 3.1.1

Factor $x - 2$ out of $(x - 2) (x^{2} - 4) (x + 2)$ .

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

Step 3.1.2

Cancel the common factor.

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x^{2} - 4$ .

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Step 3.2.1

Move the leading negative in $- \frac{3}{x^{2} - 4}$ into the numerator.

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

Step 3.2.2

Factor $x^{2} - 4$ out of $(x - 2) (x^{2} - 4) (x + 2)$ .

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $x^{2} - 4$ .

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Step 3.3.1

Factor $x^{2} - 4$ out of $(x - 2) (x^{2} - 4) (x + 2)$ .

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

Cancel the common factor of $x + 2$ .

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Step 3.4.1

Move the leading negative in $- \frac{2}{x + 2}$ into the numerator.

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

Step 3.4.2

Factor $x + 2$ out of $(x - 2) (x^{2} - 4) (x + 2)$ .

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

Step 3.4.3

Cancel the common factor.

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

Step 3.4.4

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Factor $x + 2$ out of $(x^{2} - 4) (x + 2) \cdot 2 + (x - 2) (x + 2) \cdot - 3$ .

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Step 4.1.1

Factor $x + 2$ out of $(x^{2} - 4) (x + 2) \cdot 2$ .

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

Step 4.1.2

Factor $x + 2$ out of $(x - 2) (x + 2) \cdot - 3$ .

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

Step 4.1.3

Factor $x + 2$ out of $(x + 2) ((x^{2} - 4) \cdot 2) + (x + 2) ((x - 2) \cdot - 3)$ .

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Move $2$ to the left of $x^{2}$ .

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

Step 4.4

Multiply $- 4$ by $2$ .

$\frac{(x + 2) (2 \cdot x^{2} - 8 + (x - 2) \cdot - 3)}{(x - 2) (x + 2) \cdot 5 + (x - 2) (x^{2} - 4) \cdot - 2}$

Step 4.5

Apply the distributive property.

$\frac{(x + 2) (2 x^{2} - 8 + x \cdot - 3 - 2 \cdot - 3)}{(x - 2) (x + 2) \cdot 5 + (x - 2) (x^{2} - 4) \cdot - 2}$

Step 4.6

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

$\frac{(x + 2) (2 x^{2} - 8 - 3 \cdot x - 2 \cdot - 3)}{(x - 2) (x + 2) \cdot 5 + (x - 2) (x^{2} - 4) \cdot - 2}$

Step 4.7

Multiply $- 2$ by $- 3$ .

$\frac{(x + 2) (2 x^{2} - 8 - 3 \cdot x + 6)}{(x - 2) (x + 2) \cdot 5 + (x - 2) (x^{2} - 4) \cdot - 2}$

Step 4.8

Add $- 8$ and $6$ .

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

Step 4.9

Factor by grouping.

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Step 4.9.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $b = - 3$ .

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Step 4.9.1.1

Factor $- 3$ out of $- 3 x$ .

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

Step 4.9.1.2

Rewrite $- 3$ as $1$ plus $- 4$

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

Step 4.9.1.3

Apply the distributive property.

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

Step 4.9.1.4

Multiply $x$ by $1$ .

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

Step 4.9.2

Factor out the greatest common factor from each group.

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Step 4.9.2.1

Group the first two terms and the last two terms.

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

Step 4.9.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 4.9.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

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

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

Step 5

Simplify the denominator.

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Step 5.1

Factor $x - 2$ out of $(x - 2) (x + 2) \cdot 5 + (x - 2) (x^{2} - 4) \cdot - 2$ .

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Step 5.1.1

Factor $x - 2$ out of $(x - 2) (x + 2) \cdot 5$ .

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

Step 5.1.2

Factor $x - 2$ out of $(x - 2) (x^{2} - 4) \cdot - 2$ .

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

Step 5.1.3

Factor $x - 2$ out of $(x - 2) ((x + 2) \cdot 5) + (x - 2) ((x^{2} - 4) \cdot - 2)$ .

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Move $5$ to the left of $x$ .

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

Step 5.4

Multiply $2$ by $5$ .

$\frac{(x + 2) (2 x + 1) (x - 2)}{(x - 2) (5 \cdot x + 10 + (x^{2} - 4) \cdot - 2)}$

Step 5.5

Apply the distributive property.

$\frac{(x + 2) (2 x + 1) (x - 2)}{(x - 2) (5 x + 10 + x^{2} \cdot - 2 - 4 \cdot - 2)}$

Step 5.6

Move $- 2$ to the left of $x^{2}$ .

$\frac{(x + 2) (2 x + 1) (x - 2)}{(x - 2) (5 x + 10 - 2 \cdot x^{2} - 4 \cdot - 2)}$

Step 5.7

Multiply $- 4$ by $- 2$ .

$\frac{(x + 2) (2 x + 1) (x - 2)}{(x - 2) (5 x + 10 - 2 \cdot x^{2} + 8)}$

Step 5.8

Add $10$ and $8$ .

$\frac{(x + 2) (2 x + 1) (x - 2)}{(x - 2) (5 x - 2 x^{2} + 18)}$

Step 5.9

Reorder terms.

$\frac{(x + 2) (2 x + 1) (x - 2)}{(x - 2) (- 2 x^{2} + 5 x + 18)}$

Step 5.10

Factor by grouping.

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Step 5.10.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 2 \cdot 18 = - 36$ and whose sum is $b = 5$ .

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Step 5.10.1.1

Factor $5$ out of $5 x$ .

$\frac{(x + 2) (2 x + 1) (x - 2)}{(x - 2) (- 2 x^{2} + 5 (x) + 18)}$

Step 5.10.1.2

Rewrite $5$ as $- 4$ plus $9$

$\frac{(x + 2) (2 x + 1) (x - 2)}{(x - 2) (- 2 x^{2} + (- 4 + 9) x + 18)}$

Step 5.10.1.3

Apply the distributive property.

$\frac{(x + 2) (2 x + 1) (x - 2)}{(x - 2) (- 2 x^{2} - 4 x + 9 x + 18)}$

Step 5.10.2

Factor out the greatest common factor from each group.

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Step 5.10.2.1

Group the first two terms and the last two terms.

$\frac{(x + 2) (2 x + 1) (x - 2)}{(x - 2) ((- 2 x^{2} - 4 x) + 9 x + 18)}$

Step 5.10.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 5.10.3

Factor the polynomial by factoring out the greatest common factor, $- x - 2$ .

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

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

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $x - 2$ .

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Step 6.1.1

Cancel the common factor.

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

Step 6.1.2

Rewrite the expression.

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

Step 6.2

Cancel the common factor of $x + 2$ and $- x - 2$ .

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Step 6.2.1

Factor $- 1$ out of $x$ .

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

Step 6.2.2

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

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

Step 6.2.3

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

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

Step 6.2.4

Cancel the common factor.

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

Step 6.2.5

Rewrite the expression.

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

Step 6.3

Move the negative in front of the fraction.

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

Step 7

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$2 x - 9$

Step 8

Remove parentheses.

$2 x - 9$