Algebra Examples

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

Step 1

Subtract $4$ from both sides of the equation.

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

Step 2

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

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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 $x^{2}$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

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

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

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

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

Step 2.3.2

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Cancel the common factor of $x - 2$ .

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Step 2.5.1

Cancel the common factor.

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

Step 2.5.2

Rewrite the expression.

$\frac{1}{x} - 4 = 0$

Step 3

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

$\frac{1}{x} - 4 \cdot \frac{x}{x} = 0$

Step 4

Combine $- 4$ and $\frac{x}{x}$ .

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

Step 5

Combine the numerators over the common denominator.

$\frac{1 - 4 x}{x} = 0$

Step 6

Set the numerator equal to zero.

$1 - 4 x = 0$

Step 7

Solve the equation for $x$ .

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Step 7.1

Subtract $1$ from both sides of the equation.

$- 4 x = - 1$

Step 7.2

Divide each term in $- 4 x = - 1$ by $- 4$ and simplify.

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Step 7.2.1

Divide each term in $- 4 x = - 1$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 1}{- 4}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $- 4$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 1}{- 4}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 4}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{1}{4}$