Algebra Examples

$\frac{2 x}{x - 2} - \frac{11}{x} = \frac{8}{x^{2} - 2 x}$

Step 1

Subtract $\frac{8}{x^{2} - 2 x}$ from both sides of the equation.

$\frac{2 x}{x - 2} - \frac{11}{x} - \frac{8}{x^{2} - 2 x} = 0$

Step 2

Simplify $\frac{2 x}{x - 2} - \frac{11}{x} - \frac{8}{x^{2} - 2 x}$ .

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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 $x^{2}$ .

$\frac{2 x}{x - 2} - \frac{11}{x} - \frac{8}{x \cdot x - 2 x} = 0$

Step 2.1.2

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

$\frac{2 x}{x - 2} - \frac{11}{x} - \frac{8}{x \cdot x + x \cdot - 2} = 0$

Step 2.1.3

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

$\frac{2 x}{x - 2} - \frac{11}{x} - \frac{8}{x (x - 2)} = 0$

Step 2.2

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

$\frac{2 x}{x - 2} \cdot \frac{x}{x} - \frac{11}{x} - \frac{8}{x (x - 2)} = 0$

Step 2.3

To write $- \frac{11}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{2 x}{x - 2} \cdot \frac{x}{x} - \frac{11}{x} \cdot \frac{x - 2}{x - 2} - \frac{8}{x (x - 2)} = 0$

Step 2.4

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 2.4.1

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

$\frac{2 x \cdot x}{(x - 2) x} - \frac{11}{x} \cdot \frac{x - 2}{x - 2} - \frac{8}{x (x - 2)} = 0$

Step 2.4.2

Multiply $\frac{11}{x}$ by $\frac{x - 2}{x - 2}$ .

$\frac{2 x \cdot x}{(x - 2) x} - \frac{11 (x - 2)}{x (x - 2)} - \frac{8}{x (x - 2)} = 0$

Step 2.4.3

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

$\frac{2 x \cdot x}{x (x - 2)} - \frac{11 (x - 2)}{x (x - 2)} - \frac{8}{x (x - 2)} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{2 x \cdot x - 11 (x - 2)}{x (x - 2)} - \frac{8}{x (x - 2)} = 0$

Step 2.6

Combine the numerators over the common denominator.

$\frac{2 x \cdot x - 11 (x - 2) - 8}{x (x - 2)} = 0$

Step 2.7

Simplify each term.

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Step 2.7.1

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

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Step 2.7.1.1

Move $x$ .

$\frac{2 (x \cdot x) - 11 (x - 2) - 8}{x (x - 2)} = 0$

Step 2.7.1.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} - 11 (x - 2) - 8}{x (x - 2)} = 0$

Step 2.7.2

Apply the distributive property.

$\frac{2 x^{2} - 11 x - 11 \cdot - 2 - 8}{x (x - 2)} = 0$

Step 2.7.3

Multiply $- 11$ by $- 2$ .

$\frac{2 x^{2} - 11 x + 22 - 8}{x (x - 2)} = 0$

Step 2.8

Subtract $8$ from $22$ .

$\frac{2 x^{2} - 11 x + 14}{x (x - 2)} = 0$

Step 2.9

Factor by grouping.

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Step 2.9.1

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 $a \cdot c = 2 \cdot 14 = 28$ and whose sum is $b = - 11$ .

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Step 2.9.1.1

Factor $- 11$ out of $- 11 x$ .

$\frac{2 x^{2} - 11 (x) + 14}{x (x - 2)} = 0$

Step 2.9.1.2

Rewrite $- 11$ as $- 4$ plus $- 7$

$\frac{2 x^{2} + (- 4 - 7) x + 14}{x (x - 2)} = 0$

Step 2.9.1.3

Apply the distributive property.

$\frac{2 x^{2} - 4 x - 7 x + 14}{x (x - 2)} = 0$

Step 2.9.2

Factor out the greatest common factor from each group.

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Step 2.9.2.1

Group the first two terms and the last two terms.

$\frac{(2 x^{2} - 4 x) - 7 x + 14}{x (x - 2)} = 0$

Step 2.9.2.2

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

$\frac{2 x (x - 2) - 7 (x - 2)}{x (x - 2)} = 0$

Step 2.9.3

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

$\frac{(x - 2) (2 x - 7)}{x (x - 2)} = 0$

Step 2.10

Cancel the common factor of $x - 2$ .

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Step 2.10.1

Cancel the common factor.

$\frac{(x - 2) (2 x - 7)}{x (x - 2)} = 0$

Step 2.10.2

Rewrite the expression.

$\frac{2 x - 7}{x} = 0$

Step 3

Set the denominator in $\frac{2 x - 7}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 4