Algebra Examples

$x - \frac{1}{x} = 5$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x, 1$

Step 1.2

The LCM of one and any expression is the expression.

$x$

Step 2

Multiply each term in $x - \frac{1}{x} = 5$ by $x$ to eliminate the fractions.

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Step 2.1

Multiply each term in $x - \frac{1}{x} = 5$ by $x$ .

$x \cdot x - \frac{1}{x} x = 5 x$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $x$ by $x$ .

$x^{2} - \frac{1}{x} x = 5 x$

Step 2.2.1.2

Cancel the common factor of $x$ .

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Step 2.2.1.2.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

$x^{2} + \frac{- 1}{x} x = 5 x$

Step 2.2.1.2.2

Cancel the common factor.

$x^{2} + \frac{- 1}{x} x = 5 x$

Step 2.2.1.2.3

Rewrite the expression.

$x^{2} - 1 = 5 x$

Step 3

Solve the equation.

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Step 3.1

Subtract $5 x$ from both sides of the equation.

$x^{2} - 1 - 5 x = 0$

Step 3.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.3

Substitute the values $a = 1$ , $b = - 5$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (1 \cdot - 1)}}{2 \cdot 1}$

Step 3.4

Simplify.

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Step 3.4.1

Simplify the numerator.

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Step 3.4.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 3.4.1.2

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 3.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot - 1}}{2 \cdot 1}$

Step 3.4.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{5 \pm \sqrt{25 + 4}}{2 \cdot 1}$

Step 3.4.1.3

Add $25$ and $4$ .

$x = \frac{5 \pm \sqrt{29}}{2 \cdot 1}$

Step 3.4.2

Multiply $2$ by $1$ .

$x = \frac{5 \pm \sqrt{29}}{2}$

Step 3.5

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{29}}{2}, \frac{5 - \sqrt{29}}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 + \sqrt{29}}{2}, \frac{5 - \sqrt{29}}{2}$

Decimal Form:

$x = 5.19258240 \dots, - 0.19258240 \dots$