Algebra Examples

$\frac{3 x}{x + 2} + \frac{6}{x} = \frac{12}{x^{2} + 2 x}$

Step 1

Subtract $\frac{12}{x^{2} + 2 x}$ from both sides of the equation.

$\frac{3 x}{x + 2} + \frac{6}{x} - \frac{12}{x^{2} + 2 x} = 0$

Step 2

Simplify $\frac{3 x}{x + 2} + \frac{6}{x} - \frac{12}{x^{2} + 2 x}$ .

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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 $x^{2}$ .

$\frac{3 x}{x + 2} + \frac{6}{x} - \frac{12}{x \cdot x + 2 x} = 0$

Step 2.1.2

Factor $x$ out of $2 x$ .

$\frac{3 x}{x + 2} + \frac{6}{x} - \frac{12}{x \cdot x + x \cdot 2} = 0$

Step 2.1.3

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

$\frac{3 x}{x + 2} + \frac{6}{x} - \frac{12}{x (x + 2)} = 0$

Step 2.2

To write $\frac{3 x}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{3 x}{x + 2} \cdot \frac{x}{x} + \frac{6}{x} - \frac{12}{x (x + 2)} = 0$

Step 2.3

To write $\frac{6}{x}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{3 x}{x + 2} \cdot \frac{x}{x} + \frac{6}{x} \cdot \frac{x + 2}{x + 2} - \frac{12}{x (x + 2)} = 0$

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 $\frac{3 x}{x + 2}$ by $\frac{x}{x}$ .

$\frac{3 x \cdot x}{(x + 2) x} + \frac{6}{x} \cdot \frac{x + 2}{x + 2} - \frac{12}{x (x + 2)} = 0$

Step 2.4.2

Multiply $\frac{6}{x}$ by $\frac{x + 2}{x + 2}$ .

$\frac{3 x \cdot x}{(x + 2) x} + \frac{6 (x + 2)}{x (x + 2)} - \frac{12}{x (x + 2)} = 0$

Step 2.4.3

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

$\frac{3 x \cdot x}{x (x + 2)} + \frac{6 (x + 2)}{x (x + 2)} - \frac{12}{x (x + 2)} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{3 x \cdot x + 6 (x + 2)}{x (x + 2)} - \frac{12}{x (x + 2)} = 0$

Step 2.6

Combine the numerators over the common denominator.

$\frac{3 x \cdot x + 6 (x + 2) - 12}{x (x + 2)} = 0$

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 $x$ .

$\frac{3 (x \cdot x) + 6 (x + 2) - 12}{x (x + 2)} = 0$

Step 2.7.1.2

Multiply $x$ by $x$ .

$\frac{3 x^{2} + 6 (x + 2) - 12}{x (x + 2)} = 0$

Step 2.7.2

Apply the distributive property.

$\frac{3 x^{2} + 6 x + 6 \cdot 2 - 12}{x (x + 2)} = 0$

Step 2.7.3

Multiply $6$ by $2$ .

$\frac{3 x^{2} + 6 x + 12 - 12}{x (x + 2)} = 0$

Step 2.8

Combine the opposite terms in $3 x^{2} + 6 x + 12 - 12$ .

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Step 2.8.1

Subtract $12$ from $12$ .

$\frac{3 x^{2} + 6 x + 0}{x (x + 2)} = 0$

Step 2.8.2

Add $3 x^{2} + 6 x$ and $0$ .

$\frac{3 x^{2} + 6 x}{x (x + 2)} = 0$

Step 2.9

Factor $3 x$ out of $3 x^{2} + 6 x$ .

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Step 2.9.1

Factor $3 x$ out of $3 x^{2}$ .

$\frac{3 x (x) + 6 x}{x (x + 2)} = 0$

Step 2.9.2

Factor $3 x$ out of $6 x$ .

$\frac{3 x (x) + 3 x (2)}{x (x + 2)} = 0$

Step 2.9.3

Factor $3 x$ out of $3 x (x) + 3 x (2)$ .

$\frac{3 x (x + 2)}{x (x + 2)} = 0$

Step 2.10

Cancel the common factor of $x$ .

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Step 2.10.1

Cancel the common factor.

$\frac{3 x (x + 2)}{x (x + 2)} = 0$

Step 2.10.2

Rewrite the expression.

$\frac{3 (x + 2)}{x + 2} = 0$

Step 2.11

Cancel the common factor of $x + 2$ .

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Step 2.11.1

Cancel the common factor.

$\frac{3 (x + 2)}{x + 2} = 0$

Step 2.11.2

Divide $3$ by $1$ .

$3 = 0$

Step 3

Since $3 \neq 0$ , there are no solutions.

No solution