Solve for x (1-6x^2)/x=1/x-8x^2

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

Find the LCD of the terms in the equation.

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

$x, x, 1$

Since $x, x, 1$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1$ then find LCM for the variable part $x^{1}, x^{1}$ .

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

The factor for $x^{1}$ is $x$ itself.

$x^{1} = x$

$x$ occurs $1$ time.

The LCM of $x^{1}, x^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x$

Multiply each term in $\frac{1 - 6 x^{2}}{x} = \frac{1}{x} - 8 x^{2}$ by $x$ to eliminate the fractions.

Multiply each term in $\frac{1 - 6 x^{2}}{x} = \frac{1}{x} - 8 x^{2}$ by $x$ .

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

Simplify the left side.

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the right side.

Simplify each term.

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $x^{2}$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x^{2}$ .

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

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $2$ .

$1 - 6 x^{2} = 1 - 8 x^{3}$

Solve the equation.

Add $8 x^{3}$ to both sides of the equation.

$1 - 6 x^{2} + 8 x^{3} = 1$

Subtract $1$ from both sides of the equation.

$1 - 6 x^{2} + 8 x^{3} - 1 = 0$

Combine the opposite terms in $1 - 6 x^{2} + 8 x^{3} - 1$ .

Subtract $1$ from $1$ .

$- 6 x^{2} + 8 x^{3} + 0 = 0$

Add $- 6 x^{2} + 8 x^{3}$ and $0$ .

$- 6 x^{2} + 8 x^{3} = 0$

Factor $- 2 x^{2}$ out of $- 6 x^{2} + 8 x^{3}$ .

Reorder $- 6 x^{2}$ and $8 x^{3}$ .

$8 x^{3} - 6 x^{2} = 0$

Factor $- 2 x^{2}$ out of $8 x^{3}$ .

$- 2 x^{2} (- 4 x) - 6 x^{2} = 0$

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

$- 2 x^{2} (- 4 x) - 2 x^{2} \cdot 3 = 0$

Factor $- 2 x^{2}$ out of $- 2 x^{2} (- 4 x) - 2 x^{2} (3)$ .

$- 2 x^{2} (- 4 x + 3) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} = 0$

$- 4 x + 3 = 0$

Set $x^{2}$ equal to $0$ and solve for $x$ .

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Solve $x^{2} = 0$ for $x$ .

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

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

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

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Set $- 4 x + 3$ equal to $0$ and solve for $x$ .

Set $- 4 x + 3$ equal to $0$ .

$- 4 x + 3 = 0$

Solve $- 4 x + 3 = 0$ for $x$ .

Subtract $3$ from both sides of the equation.

$- 4 x = - 3$

Divide each term in $- 4 x = - 3$ by $- 4$ and simplify.

Divide each term in $- 4 x = - 3$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 3}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 3}{- 4}$

Divide $x$ by $1$ .

$x = \frac{- 3}{- 4}$

Simplify the right side.

Dividing two negative values results in a positive value.

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

The final solution is all the values that make $- 2 x^{2} (- 4 x + 3) = 0$ true.

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

Exclude the solutions that do not make $\frac{1 - 6 x^{2}}{x} = \frac{1}{x} - 8 x^{2}$ true.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.75$

Solve for x (1-6x^2)/x=1/x-8x^2

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