Pre-Algebra Examples

$- \frac{1}{x} + 2 = 2 x - 4$

Step 1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.1

Subtract $2$ from both sides of the equation.

$- \frac{1}{x} = 2 x - 4 - 2$

Step 1.2

Subtract $2$ from $- 4$ .

$- \frac{1}{x} = 2 x - 6$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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

$x, 1, 1$

Step 2.2

The LCM of one and any expression is the expression.

$x$

Step 3

Multiply each term in $- \frac{1}{x} = 2 x - 6$ by $x$ to eliminate the fractions.

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Step 3.1

Multiply each term in $- \frac{1}{x} = 2 x - 6$ by $x$ .

$- \frac{1}{x} x = 2 x \cdot x - 6 x$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $x$ .

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Step 3.2.1.1

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

$\frac{- 1}{x} x = 2 x \cdot x - 6 x$

Step 3.2.1.2

Cancel the common factor.

$\frac{- 1}{x} x = 2 x \cdot x - 6 x$

Step 3.2.1.3

Rewrite the expression.

$- 1 = 2 x \cdot x - 6 x$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

Move $x$ .

$- 1 = 2 (x \cdot x) - 6 x$

Step 3.3.1.2

Multiply $x$ by $x$ .

$- 1 = 2 x^{2} - 6 x$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $2 x^{2} - 6 x = - 1$ .

$2 x^{2} - 6 x = - 1$

Step 4.2

Add $1$ to both sides of the equation.

$2 x^{2} - 6 x + 1 = 0$

Step 4.3

Use the quadratic formula to find the solutions.

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

Step 4.4

Substitute the values $a = 2$ , $b = - 6$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (2 \cdot 1)}}{2 \cdot 2}$

Step 4.5

Simplify.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Step 4.5.1.2

Multiply $- 4 \cdot 2 \cdot 1$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{6 \pm \sqrt{36 - 8 \cdot 1}}{2 \cdot 2}$

Step 4.5.1.2.2

Multiply $- 8$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 8}}{2 \cdot 2}$

Step 4.5.1.3

Subtract $8$ from $36$ .

$x = \frac{6 \pm \sqrt{28}}{2 \cdot 2}$

Step 4.5.1.4

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 4.5.1.4.1

Factor $4$ out of $28$ .

$x = \frac{6 \pm \sqrt{4 (7)}}{2 \cdot 2}$

Step 4.5.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{6 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot 2}$

Step 4.5.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{7}}{2 \cdot 2}$

Step 4.5.2

Multiply $2$ by $2$ .

$x = \frac{6 \pm 2 \sqrt{7}}{4}$

Step 4.5.3

Simplify $\frac{6 \pm 2 \sqrt{7}}{4}$ .

$x = \frac{3 \pm \sqrt{7}}{2}$

Step 4.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 4.6.1

Simplify the numerator.

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Step 4.6.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Step 4.6.1.2

Multiply $- 4 \cdot 2 \cdot 1$ .

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Step 4.6.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{6 \pm \sqrt{36 - 8 \cdot 1}}{2 \cdot 2}$

Step 4.6.1.2.2

Multiply $- 8$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 8}}{2 \cdot 2}$

Step 4.6.1.3

Subtract $8$ from $36$ .

$x = \frac{6 \pm \sqrt{28}}{2 \cdot 2}$

Step 4.6.1.4

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 4.6.1.4.1

Factor $4$ out of $28$ .

$x = \frac{6 \pm \sqrt{4 (7)}}{2 \cdot 2}$

Step 4.6.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{6 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot 2}$

Step 4.6.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{7}}{2 \cdot 2}$

Step 4.6.2

Multiply $2$ by $2$ .

$x = \frac{6 \pm 2 \sqrt{7}}{4}$

Step 4.6.3

Simplify $\frac{6 \pm 2 \sqrt{7}}{4}$ .

$x = \frac{3 \pm \sqrt{7}}{2}$

Step 4.6.4

Change the $\pm$ to $+$ .

$x = \frac{3 + \sqrt{7}}{2}$

Step 4.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 4.7.1

Simplify the numerator.

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Step 4.7.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Step 4.7.1.2

Multiply $- 4 \cdot 2 \cdot 1$ .

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Step 4.7.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{6 \pm \sqrt{36 - 8 \cdot 1}}{2 \cdot 2}$

Step 4.7.1.2.2

Multiply $- 8$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 8}}{2 \cdot 2}$

Step 4.7.1.3

Subtract $8$ from $36$ .

$x = \frac{6 \pm \sqrt{28}}{2 \cdot 2}$

Step 4.7.1.4

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 4.7.1.4.1

Factor $4$ out of $28$ .

$x = \frac{6 \pm \sqrt{4 (7)}}{2 \cdot 2}$

Step 4.7.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{6 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot 2}$

Step 4.7.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{7}}{2 \cdot 2}$

Step 4.7.2

Multiply $2$ by $2$ .

$x = \frac{6 \pm 2 \sqrt{7}}{4}$

Step 4.7.3

Simplify $\frac{6 \pm 2 \sqrt{7}}{4}$ .

$x = \frac{3 \pm \sqrt{7}}{2}$

Step 4.7.4

Change the $\pm$ to $-$ .

$x = \frac{3 - \sqrt{7}}{2}$

Step 4.8

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{7}}{2}, \frac{3 - \sqrt{7}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 + \sqrt{7}}{2}, \frac{3 - \sqrt{7}}{2}$

Decimal Form:

$x = 2.82287565 \dots, 0.17712434 \dots$