Algebra Examples

$\frac{1 - 6 x^{2}}{x} = \frac{1}{x} - 8 x^{2}$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify $\frac{1 - 6 x^{2}}{x}$ .

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Step 1.1.1.1

Split the fraction $\frac{1 - 6 x^{2}}{x}$ into two fractions.

$\frac{1}{x} + \frac{- 6 x^{2}}{x} = \frac{1}{x} - 8 x^{2}$

Step 1.1.1.2

Cancel the common factor of $x^{2}$ and $x$ .

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Step 1.1.1.2.1

Factor $x$ out of $- 6 x^{2}$ .

$\frac{1}{x} + \frac{x (- 6 x)}{x} = \frac{1}{x} - 8 x^{2}$

Step 1.1.1.2.2

Cancel the common factors.

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Step 1.1.1.2.2.1

Raise $x$ to the power of $1$ .

$\frac{1}{x} + \frac{x (- 6 x)}{x^{1}} = \frac{1}{x} - 8 x^{2}$

Step 1.1.1.2.2.2

Factor $x$ out of $x^{1}$ .

$\frac{1}{x} + \frac{x (- 6 x)}{x \cdot 1} = \frac{1}{x} - 8 x^{2}$

Step 1.1.1.2.2.3

Cancel the common factor.

$\frac{1}{x} + \frac{x (- 6 x)}{x \cdot 1} = \frac{1}{x} - 8 x^{2}$

Step 1.1.1.2.2.4

Rewrite the expression.

$\frac{1}{x} + \frac{- 6 x}{1} = \frac{1}{x} - 8 x^{2}$

Step 1.1.1.2.2.5

Divide $- 6 x$ by $1$ .

$\frac{1}{x} - 6 x = \frac{1}{x} - 8 x^{2}$

Step 1.2

Move all the expressions to the left side of the equation.

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Step 1.2.1

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

$\frac{1}{x} - 6 x - \frac{1}{x} = - 8 x^{2}$

Step 1.2.2

Add $8 x^{2}$ to both sides of the equation.

$\frac{1}{x} - 6 x - \frac{1}{x} + 8 x^{2} = 0$

Step 1.3

Combine the opposite terms in $\frac{1}{x} - 6 x - \frac{1}{x} + 8 x^{2}$ .

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Step 1.3.1

Subtract $\frac{1}{x}$ from $\frac{1}{x}$ .

$- 6 x + 0 + 8 x^{2} = 0$

Step 1.3.2

Add $- 6 x$ and $0$ .

$- 6 x + 8 x^{2} = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 8$ , $b = - 6$ , and $c = 0$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (8 \cdot 0)}}{2 \cdot 8}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 8 \cdot 0}}{2 \cdot 8}$

Step 4.1.2

Multiply $- 4 \cdot 8 \cdot 0$ .

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Step 4.1.2.1

Multiply $- 4$ by $8$ .

$x = \frac{6 \pm \sqrt{36 - 32 \cdot 0}}{2 \cdot 8}$

Step 4.1.2.2

Multiply $- 32$ by $0$ .

$x = \frac{6 \pm \sqrt{36 + 0}}{2 \cdot 8}$

Step 4.1.3

Add $36$ and $0$ .

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

Step 4.1.4

Rewrite $36$ as $6^{2}$ .

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

Step 4.1.5

Pull terms out from under the radical, assuming positive real numbers.

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

Step 4.2

Multiply $2$ by $8$ .

$x = \frac{6 \pm 6}{16}$

Step 4.3

Simplify $\frac{6 \pm 6}{16}$ .

$x = \frac{3 \pm 3}{8}$

Step 5

The final answer is the combination of both solutions.

$x = \frac{3}{4}, 0$