Algebra Examples

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

Step 1

Subtract $2$ from both sides of the equation.

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

Step 2

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

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Step 2.1

Factor $x$ out of $x^{2} - 2 x$ .

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Step 2.1.1

Factor $x$ out of $x^{2}$ .

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

Step 2.1.2

Factor $x$ out of $- 2 x$ .

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

Step 2.1.3

Factor $x$ out of $x \cdot x + x \cdot - 2$ .

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify each term.

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Step 2.5.1

Factor $2$ out of $2 x - 4$ .

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Step 2.5.1.1

Factor $2$ out of $2 x$ .

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

Step 2.5.1.2

Factor $2$ out of $- 4$ .

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

Step 2.5.1.3

Factor $2$ out of $2 x + 2 \cdot - 2$ .

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

Step 2.5.2

Cancel the common factor of $x - 2$ .

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Step 2.5.2.1

Cancel the common factor.

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

Step 2.5.2.2

Rewrite the expression.

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

Step 3

Factor $2$ out of $\frac{2}{x} - 2$ .

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Step 3.1

Factor $2$ out of $\frac{2}{x}$ .

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

Step 3.2

Factor $2$ out of $- 2$ .

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

Step 3.3

Factor $2$ out of $2 \frac{1}{x} + 2 \cdot - 1$ .

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

Step 4

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

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

Step 5

Combine $- 1$ and $\frac{x}{x}$ .

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Combine $2$ and $\frac{1 - x}{x}$ .

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

Step 8

Set the numerator equal to zero.

$2 (1 - x) = 0$

Step 9

Solve the equation for $x$ .

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Step 9.1

Divide each term in $2 (1 - x) = 0$ by $2$ and simplify.

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Step 9.1.1

Divide each term in $2 (1 - x) = 0$ by $2$ .

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

Step 9.1.2

Simplify the left side.

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Step 9.1.2.1

Cancel the common factor of $2$ .

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Step 9.1.2.1.1

Cancel the common factor.

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

Step 9.1.2.1.2

Divide $1 - x$ by $1$ .

$1 - x = \frac{0}{2}$

Step 9.1.3

Simplify the right side.

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Step 9.1.3.1

Divide $0$ by $2$ .

$1 - x = 0$

Step 9.2

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 9.3

Divide each term in $- x = - 1$ by $- 1$ and simplify.

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Step 9.3.1

Divide each term in $- x = - 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 1}{- 1}$

Step 9.3.2

Simplify the left side.

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Step 9.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 1}{- 1}$

Step 9.3.2.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 1}$

Step 9.3.3

Simplify the right side.

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Step 9.3.3.1

Divide $- 1$ by $- 1$ .

$x = 1$