Algebra Examples

$\frac{1}{x} - \frac{x}{6} = \frac{2}{3}$

Step 1

Subtract $\frac{2}{3}$ from both sides of the equation.

$\frac{1}{x} - \frac{x}{6} - \frac{2}{3} = 0$

Step 2

Reorder terms.

$- \frac{x}{6} + \frac{1}{x} - \frac{2}{3} = 0$

Step 3

To write $- \frac{x}{6}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$- \frac{x}{6} \cdot \frac{x}{x} + \frac{1}{x} - \frac{2}{3} = 0$

Step 4

To write $\frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{x}{6} \cdot \frac{x}{x} + \frac{1}{x} \cdot \frac{6}{6} - \frac{2}{3} = 0$

Step 5

Write each expression with a common denominator of $6 x$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1

Multiply $\frac{x}{6}$ by $\frac{x}{x}$ .

$- \frac{x \cdot x}{6 x} + \frac{1}{x} \cdot \frac{6}{6} - \frac{2}{3} = 0$

Step 5.2

Multiply $\frac{1}{x}$ by $\frac{6}{6}$ .

$- \frac{x \cdot x}{6 x} + \frac{6}{x \cdot 6} - \frac{2}{3} = 0$

Step 5.3

Reorder the factors of $x \cdot 6$ .

$- \frac{x \cdot x}{6 x} + \frac{6}{6 x} - \frac{2}{3} = 0$

Step 6

Combine the numerators over the common denominator.

$\frac{- x \cdot x + 6}{6 x} - \frac{2}{3} = 0$

Step 7

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

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Step 7.1

Move $x$ .

$\frac{- (x \cdot x) + 6}{6 x} - \frac{2}{3} = 0$

Step 7.2

Multiply $x$ by $x$ .

$\frac{- x^{2} + 6}{6 x} - \frac{2}{3} = 0$

Step 8

To write $- \frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{2 x}{2 x}$ .

$\frac{- x^{2} + 6}{6 x} - \frac{2}{3} \cdot \frac{2 x}{2 x} = 0$

Step 9

Write each expression with a common denominator of $6 x$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.1

Multiply $\frac{2}{3}$ by $\frac{2 x}{2 x}$ .

$\frac{- x^{2} + 6}{6 x} - \frac{2 (2 x)}{3 (2 x)} = 0$

Step 9.2

Multiply $2$ by $3$ .

$\frac{- x^{2} + 6}{6 x} - \frac{2 (2 x)}{6 x} = 0$

Step 10

Combine the numerators over the common denominator.

$\frac{- x^{2} + 6 - 2 (2 x)}{6 x} = 0$

Step 11

Remove parentheses.

$\frac{- x^{2} + 6 - 2 \cdot (2 x)}{6 x} = 0$

Step 12

Simplify the numerator.

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Step 12.1

Multiply $- 2$ by $2$ .

$\frac{- x^{2} + 6 - 4 x}{6 x} = 0$

Step 12.2

Reorder terms.

$\frac{- x^{2} - 4 x + 6}{6 x} = 0$

Step 13

Set the numerator equal to zero.

$- x^{2} - 4 x + 6 = 0$

Step 14

Solve the equation for $x$ .

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Step 14.1

Use the quadratic formula to find the solutions.

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

Step 14.2

Substitute the values $a = - 1$ , $b = - 4$ , and $c = 6$ into the quadratic formula and solve for $x$ .

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

Step 14.3

Simplify.

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Step 14.3.1

Simplify the numerator.

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Step 14.3.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 1 \cdot 6}}{2 \cdot - 1}$

Step 14.3.1.2

Multiply $- 4 \cdot - 1 \cdot 6$ .

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Step 14.3.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{4 \pm \sqrt{16 + 4 \cdot 6}}{2 \cdot - 1}$

Step 14.3.1.2.2

Multiply $4$ by $6$ .

$x = \frac{4 \pm \sqrt{16 + 24}}{2 \cdot - 1}$

Step 14.3.1.3

Add $16$ and $24$ .

$x = \frac{4 \pm \sqrt{40}}{2 \cdot - 1}$

Step 14.3.1.4

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 14.3.1.4.1

Factor $4$ out of $40$ .

$x = \frac{4 \pm \sqrt{4 (10)}}{2 \cdot - 1}$

Step 14.3.1.4.2

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot - 1}$

Step 14.3.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{10}}{2 \cdot - 1}$

Step 14.3.2

Multiply $2$ by $- 1$ .

$x = \frac{4 \pm 2 \sqrt{10}}{- 2}$

Step 14.3.3

Simplify $\frac{4 \pm 2 \sqrt{10}}{- 2}$ .

$x = \frac{2 \pm \sqrt{10}}{- 1}$

Step 14.3.4

Move the negative one from the denominator of $\frac{2 \pm \sqrt{10}}{- 1}$ .

$x = - 1 \cdot (2 \pm \sqrt{10})$

Step 14.3.5

Rewrite $- 1 \cdot (2 \pm \sqrt{10})$ as $- (2 \pm \sqrt{10})$ .

$x = - (2 \pm \sqrt{10})$

Step 14.4

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

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Step 14.4.1

Simplify the numerator.

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Step 14.4.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 1 \cdot 6}}{2 \cdot - 1}$

Step 14.4.1.2

Multiply $- 4 \cdot - 1 \cdot 6$ .

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Step 14.4.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{4 \pm \sqrt{16 + 4 \cdot 6}}{2 \cdot - 1}$

Step 14.4.1.2.2

Multiply $4$ by $6$ .

$x = \frac{4 \pm \sqrt{16 + 24}}{2 \cdot - 1}$

Step 14.4.1.3

Add $16$ and $24$ .

$x = \frac{4 \pm \sqrt{40}}{2 \cdot - 1}$

Step 14.4.1.4

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 14.4.1.4.1

Factor $4$ out of $40$ .

$x = \frac{4 \pm \sqrt{4 (10)}}{2 \cdot - 1}$

Step 14.4.1.4.2

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot - 1}$

Step 14.4.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{10}}{2 \cdot - 1}$

Step 14.4.2

Multiply $2$ by $- 1$ .

$x = \frac{4 \pm 2 \sqrt{10}}{- 2}$

Step 14.4.3

Simplify $\frac{4 \pm 2 \sqrt{10}}{- 2}$ .

$x = \frac{2 \pm \sqrt{10}}{- 1}$

Step 14.4.4

Move the negative one from the denominator of $\frac{2 \pm \sqrt{10}}{- 1}$ .

$x = - 1 \cdot (2 \pm \sqrt{10})$

Step 14.4.5

Rewrite $- 1 \cdot (2 \pm \sqrt{10})$ as $- (2 \pm \sqrt{10})$ .

$x = - (2 \pm \sqrt{10})$

Step 14.4.6

Change the $\pm$ to $+$ .

$x = - (2 + \sqrt{10})$

Step 14.4.7

Apply the distributive property.

$x = - 1 \cdot 2 - \sqrt{10}$

Step 14.4.8

Multiply $- 1$ by $2$ .

$x = - 2 - \sqrt{10}$

Step 14.5

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

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Step 14.5.1

Simplify the numerator.

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Step 14.5.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 1 \cdot 6}}{2 \cdot - 1}$

Step 14.5.1.2

Multiply $- 4 \cdot - 1 \cdot 6$ .

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Step 14.5.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{4 \pm \sqrt{16 + 4 \cdot 6}}{2 \cdot - 1}$

Step 14.5.1.2.2

Multiply $4$ by $6$ .

$x = \frac{4 \pm \sqrt{16 + 24}}{2 \cdot - 1}$

Step 14.5.1.3

Add $16$ and $24$ .

$x = \frac{4 \pm \sqrt{40}}{2 \cdot - 1}$

Step 14.5.1.4

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 14.5.1.4.1

Factor $4$ out of $40$ .

$x = \frac{4 \pm \sqrt{4 (10)}}{2 \cdot - 1}$

Step 14.5.1.4.2

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot - 1}$

Step 14.5.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{10}}{2 \cdot - 1}$

Step 14.5.2

Multiply $2$ by $- 1$ .

$x = \frac{4 \pm 2 \sqrt{10}}{- 2}$

Step 14.5.3

Simplify $\frac{4 \pm 2 \sqrt{10}}{- 2}$ .

$x = \frac{2 \pm \sqrt{10}}{- 1}$

Step 14.5.4

Move the negative one from the denominator of $\frac{2 \pm \sqrt{10}}{- 1}$ .

$x = - 1 \cdot (2 \pm \sqrt{10})$

Step 14.5.5

Rewrite $- 1 \cdot (2 \pm \sqrt{10})$ as $- (2 \pm \sqrt{10})$ .

$x = - (2 \pm \sqrt{10})$

Step 14.5.6

Change the $\pm$ to $-$ .

$x = - (2 - \sqrt{10})$

Step 14.5.7

Apply the distributive property.

$x = - 1 \cdot 2 + \sqrt{10}$

Step 14.5.8

Multiply $- 1$ by $2$ .

$x = - 2 + \sqrt{10}$

Step 14.5.9

Multiply $- - \sqrt{10}$ .

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Step 14.5.9.1

Multiply $- 1$ by $- 1$ .

$x = - 2 + 1 \sqrt{10}$

Step 14.5.9.2

Multiply $\sqrt{10}$ by $1$ .

$x = - 2 + \sqrt{10}$

Step 14.6

The final answer is the combination of both solutions.

$x = - 2 - \sqrt{10}, - 2 + \sqrt{10}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$x = - 2 - \sqrt{10}, - 2 + \sqrt{10}$

Decimal Form:

$x = - 5.16227766 \dots, 1.16227766 \dots$