Algebra Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Write each expression with a common denominator of $2 x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1

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

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

Step 2.2.2

Move $2$ to the left of $x^{2}$ .

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Subtract $1$ from $2$ .

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

Step 3

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

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

Step 4

Write each expression with a common denominator of $2 x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

Rewrite using the commutative property of multiplication.

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

Step 4.3

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

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Step 4.3.1

Move $x$ .

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

Step 4.3.2

Multiply $x$ by $x$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Set the numerator equal to zero.

$2 x + 1 = 0$

Step 7

Solve the equation for $x$ .

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Step 7.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 7.2

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

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Step 7.2.1

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

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

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $2$ .

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Step 7.2.2.1.1

Cancel the common factor.

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

Step 7.2.2.1.2

Divide $x$ by $1$ .

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

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Move the negative in front of the fraction.

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