Algebra Examples

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

Step 1

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$x \cdot x = (x^{2} - 2) \cdot - 1$

Step 2

Solve the equation for $x$ .

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Step 2.1

Simplify $x \cdot x$ .

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Step 2.1.1

Rewrite.

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

Step 2.1.2

Simplify by adding zeros.

$x \cdot x = (x^{2} - 2) \cdot - 1$

Step 2.1.3

Multiply $x$ by $x$ .

$x^{2} = (x^{2} - 2) \cdot - 1$

Step 2.2

Simplify $(x^{2} - 2) \cdot - 1$ .

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Step 2.2.1

Simplify by multiplying through.

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Step 2.2.1.1

Apply the distributive property.

$x^{2} = x^{2} \cdot - 1 - 2 \cdot - 1$

Step 2.2.1.2

Simplify the expression.

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Step 2.2.1.2.1

Move $- 1$ to the left of $x^{2}$ .

$x^{2} = - 1 \cdot x^{2} - 2 \cdot - 1$

Step 2.2.1.2.2

Multiply $- 2$ by $- 1$ .

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

Step 2.2.2

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$x^{2} = - x^{2} + 2$

Step 2.3

Move all terms containing $x$ to the left side of the equation.

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Step 2.3.1

Add $x^{2}$ to both sides of the equation.

$x^{2} + x^{2} = 2$

Step 2.3.2

Add $x^{2}$ and $x^{2}$ .

$2 x^{2} = 2$

Step 2.4

Divide each term in $2 x^{2} = 2$ by $2$ and simplify.

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Step 2.4.1

Divide each term in $2 x^{2} = 2$ by $2$ .

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $2$ .

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Step 2.4.2.1.1

Cancel the common factor.

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

Step 2.4.2.1.2

Divide $x^{2}$ by $1$ .

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

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Divide $2$ by $2$ .

$x^{2} = 1$

Step 2.5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{1}$

Step 2.6

Any root of $1$ is $1$ .

$x = \pm 1$

Step 2.7

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.7.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 1$

Step 2.7.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 1$

Step 2.7.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 1, - 1$