Algebra Examples

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

Step 1

Find the LCD of the terms in the equation.

Tap for more steps...

Step 1.1

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

$x, x, 1$

Step 1.2

Since $x, x, 1$ contains 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 $x^{1}, x^{1}$ .

Step 1.3

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.

Step 1.4

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

Not prime

Step 1.5

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

$1$

Step 1.6

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

$x^{1} = x$

$x$ occurs $1$ time.

Step 1.7

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.

$x$

Step 2

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

Tap for more steps...

Step 2.1

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

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

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Rewrite the expression.

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

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Simplify each term.

Tap for more steps...

Step 2.3.1.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 2.3.1.1.1

Cancel the common factor.

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

Step 2.3.1.1.2

Rewrite the expression.

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

Step 2.3.1.2

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

Tap for more steps...

Step 2.3.1.2.1

Move $x$ .

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

Step 2.3.1.2.2

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

Tap for more steps...

Step 2.3.1.2.2.1

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

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

Step 2.3.1.2.2.2

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

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

Step 2.3.1.2.3

Add $1$ and $2$ .

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

Step 3

Solve the equation.

Tap for more steps...

Step 3.1

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

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

Step 3.2

Subtract $1$ from both sides of the equation.

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

Step 3.3

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

Tap for more steps...

Step 3.3.1

Subtract $1$ from $1$ .

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

Step 3.3.2

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

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

Step 3.4

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

Tap for more steps...

Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.4.4

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

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

Step 3.5

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

$x^{2} = 0$

$- 4 x + 3 = 0$

Step 3.6

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

Tap for more steps...

Step 3.6.1

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

$x^{2} = 0$

Step 3.6.2

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

Tap for more steps...

Step 3.6.2.1

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

$x = \pm \sqrt{0}$

Step 3.6.2.2

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 3.6.2.2.1

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

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

Step 3.6.2.2.2

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

$x = \pm 0$

Step 3.6.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 3.7

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

Tap for more steps...

Step 3.7.1

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

$- 4 x + 3 = 0$

Step 3.7.2

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

Tap for more steps...

Step 3.7.2.1

Subtract $3$ from both sides of the equation.

$- 4 x = - 3$

Step 3.7.2.2

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

Tap for more steps...

Step 3.7.2.2.1

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

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

Step 3.7.2.2.2

Simplify the left side.

Tap for more steps...

Step 3.7.2.2.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 3.7.2.2.2.1.1

Cancel the common factor.

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

Step 3.7.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.7.2.2.3

Simplify the right side.

Tap for more steps...

Step 3.7.2.2.3.1

Dividing two negative values results in a positive value.

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

Step 3.8

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

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

Step 4

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.75$