Algebra Examples

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

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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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 $x^{2}$ .

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

Step 2.1.2

Factor $x$ out of $2 x$ .

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

Step 2.1.3

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

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

Step 2.2

Write $1$ as a fraction with a common denominator.

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Simplify the numerator.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Multiply $- 1$ by $- 1$ .

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

Step 2.4.3

Subtract $x$ from $x$ .

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

Step 2.4.4

Add $0$ and $1$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Subtract $1$ from $2$ .

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

Step 7

Set the numerator equal to zero.

$x + 1 = 0$

Step 8

Subtract $1$ from both sides of the equation.

$x = - 1$