Algebra Examples

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

Step 1

Subtract $\frac{x - 1}{x - 2}$ from both sides of the equation.

$\frac{4}{x^{2} - 4} - \frac{1}{x + 2} - \frac{x - 1}{x - 2} = 0$

Step 2

Simplify $\frac{4}{x^{2} - 4} - \frac{1}{x + 2} - \frac{x - 1}{x - 2}$ .

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Step 2.1

Simplify the denominator.

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Step 2.1.1

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

$\frac{4}{x^{2} - 2^{2}} - \frac{1}{x + 2} - \frac{x - 1}{x - 2} = 0$

Step 2.1.2

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 $b = 2$ .

$\frac{4}{(x + 2) (x - 2)} - \frac{1}{x + 2} - \frac{x - 1}{x - 2} = 0$

Step 2.2

Find the common denominator.

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Step 2.2.1

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

$\frac{4}{(x + 2) (x - 2)} - (\frac{1}{x + 2} \cdot \frac{x - 2}{x - 2}) - \frac{x - 1}{x - 2} = 0$

Step 2.2.2

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

$\frac{4}{(x + 2) (x - 2)} - \frac{x - 2}{(x + 2) (x - 2)} - \frac{x - 1}{x - 2} = 0$

Step 2.2.3

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

$\frac{4}{(x + 2) (x - 2)} - \frac{x - 2}{(x + 2) (x - 2)} - (\frac{x - 1}{x - 2} \cdot \frac{x + 2}{x + 2}) = 0$

Step 2.2.4

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

$\frac{4}{(x + 2) (x - 2)} - \frac{x - 2}{(x + 2) (x - 2)} - \frac{(x - 1) (x + 2)}{(x - 2) (x + 2)} = 0$

Step 2.2.5

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

$\frac{4}{(x + 2) (x - 2)} - \frac{x - 2}{(x + 2) (x - 2)} - \frac{(x - 1) (x + 2)}{(x + 2) (x - 2)} = 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{4 - (x - 2) - (x - 1) (x + 2)}{(x + 2) (x - 2)} = 0$

Step 2.4

Simplify each term.

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Step 2.4.1

Apply the distributive property.

$\frac{4 - x - - 2 - (x - 1) (x + 2)}{(x + 2) (x - 2)} = 0$

Step 2.4.2

Multiply $- 1$ by $- 2$ .

$\frac{4 - x + 2 - (x - 1) (x + 2)}{(x + 2) (x - 2)} = 0$

Step 2.4.3

Apply the distributive property.

$\frac{4 - x + 2 + (- x - - 1) (x + 2)}{(x + 2) (x - 2)} = 0$

Step 2.4.4

Multiply $- 1$ by $- 1$ .

$\frac{4 - x + 2 + (- x + 1) (x + 2)}{(x + 2) (x - 2)} = 0$

Step 2.4.5

Expand $(- x + 1) (x + 2)$ using the FOIL Method.

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Step 2.4.5.1

Apply the distributive property.

$\frac{4 - x + 2 - x (x + 2) + 1 (x + 2)}{(x + 2) (x - 2)} = 0$

Step 2.4.5.2

Apply the distributive property.

$\frac{4 - x + 2 - x \cdot x - x \cdot 2 + 1 (x + 2)}{(x + 2) (x - 2)} = 0$

Step 2.4.5.3

Apply the distributive property.

$\frac{4 - x + 2 - x \cdot x - x \cdot 2 + 1 x + 1 \cdot 2}{(x + 2) (x - 2)} = 0$

Step 2.4.6

Simplify and combine like terms.

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Step 2.4.6.1

Simplify each term.

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Step 2.4.6.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.4.6.1.1.1

Move $x$ .

$\frac{4 - x + 2 - (x \cdot x) - x \cdot 2 + 1 x + 1 \cdot 2}{(x + 2) (x - 2)} = 0$

Step 2.4.6.1.1.2

Multiply $x$ by $x$ .

$\frac{4 - x + 2 - x^{2} - x \cdot 2 + 1 x + 1 \cdot 2}{(x + 2) (x - 2)} = 0$

Step 2.4.6.1.2

Multiply $2$ by $- 1$ .

$\frac{4 - x + 2 - x^{2} - 2 x + 1 x + 1 \cdot 2}{(x + 2) (x - 2)} = 0$

Step 2.4.6.1.3

Multiply $x$ by $1$ .

$\frac{4 - x + 2 - x^{2} - 2 x + x + 1 \cdot 2}{(x + 2) (x - 2)} = 0$

Step 2.4.6.1.4

Multiply $2$ by $1$ .

$\frac{4 - x + 2 - x^{2} - 2 x + x + 2}{(x + 2) (x - 2)} = 0$

Step 2.4.6.2

Add $- 2 x$ and $x$ .

$\frac{4 - x + 2 - x^{2} - x + 2}{(x + 2) (x - 2)} = 0$

Step 2.5

Add $4$ and $2$ .

$\frac{- x + 6 - x^{2} - x + 2}{(x + 2) (x - 2)} = 0$

Step 2.6

Subtract $x$ from $- x$ .

$\frac{- 2 x + 6 - x^{2} + 2}{(x + 2) (x - 2)} = 0$

Step 2.7

Add $6$ and $2$ .

$\frac{- 2 x - x^{2} + 8}{(x + 2) (x - 2)} = 0$

Step 2.8

Simplify the numerator.

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Step 2.8.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$\frac{- 2 u - u^{2} + 8}{(x + 2) (x - 2)} = 0$

Step 2.8.2

Factor by grouping.

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Step 2.8.2.1

Reorder terms.

$\frac{- u^{2} - 2 u + 8}{(x + 2) (x - 2)} = 0$

Step 2.8.2.2

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 $a \cdot c = - 1 \cdot 8 = - 8$ and whose sum is $b = - 2$ .

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Step 2.8.2.2.1

Factor $- 2$ out of $- 2 u$ .

$\frac{- u^{2} - 2 (u) + 8}{(x + 2) (x - 2)} = 0$

Step 2.8.2.2.2

Rewrite $- 2$ as $2$ plus $- 4$

$\frac{- u^{2} + (2 - 4) u + 8}{(x + 2) (x - 2)} = 0$

Step 2.8.2.2.3

Apply the distributive property.

$\frac{- u^{2} + 2 u - 4 u + 8}{(x + 2) (x - 2)} = 0$

Step 2.8.2.3

Factor out the greatest common factor from each group.

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Step 2.8.2.3.1

Group the first two terms and the last two terms.

$\frac{(- u^{2} + 2 u) - 4 u + 8}{(x + 2) (x - 2)} = 0$

Step 2.8.2.3.2

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

$\frac{u (- u + 2) + 4 (- u + 2)}{(x + 2) (x - 2)} = 0$

Step 2.8.2.4

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

$\frac{(- u + 2) (u + 4)}{(x + 2) (x - 2)} = 0$

Step 2.8.3

Replace all occurrences of $u$ with $x$ .

$\frac{(- x + 2) (x + 4)}{(x + 2) (x - 2)} = 0$

Step 2.9

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

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Step 2.9.1

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

$\frac{(- (x) + 2) (x + 4)}{(x + 2) (x - 2)} = 0$

Step 2.9.2

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

$\frac{(- (x) - 1 (- 2)) (x + 4)}{(x + 2) (x - 2)} = 0$

Step 2.9.3

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

$\frac{- (x - 2) (x + 4)}{(x + 2) (x - 2)} = 0$

Step 2.9.4

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

$\frac{- 1 (x - 2) (x + 4)}{(x + 2) (x - 2)} = 0$

Step 2.9.5

Cancel the common factor.

$\frac{- 1 (x - 2) (x + 4)}{(x + 2) (x - 2)} = 0$

Step 2.9.6

Rewrite the expression.

$\frac{- 1 (x + 4)}{x + 2} = 0$

Step 2.10

Move the negative in front of the fraction.

$- \frac{x + 4}{x + 2} = 0$

Step 3

Set the numerator equal to zero.

$x + 4 = 0$

Step 4

Subtract $4$ from both sides of the equation.

$x = - 4$