Algebra Examples

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

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

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

$1, x, 1$

Step 1.2

The LCM of one and any expression is the expression.

$x$

Step 2

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

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Step 2.1

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

$8 x \cdot x - \frac{1}{x} x = - 2 x$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

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

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Step 2.2.1.1.1

Move $x$ .

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

Step 2.2.1.1.2

Multiply $x$ by $x$ .

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

Step 2.2.1.2

Cancel the common factor of $x$ .

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Step 2.2.1.2.1

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

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

Step 2.2.1.2.2

Cancel the common factor.

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

Step 2.2.1.2.3

Rewrite the expression.

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

Step 3

Solve the equation.

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Step 3.1

Add $2 x$ to both sides of the equation.

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

Step 3.2

Factor by grouping.

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Step 3.2.1

Reorder terms.

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

Step 3.2.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 8 \cdot - 1 = - 8$ and whose sum is $b = 2$ .

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Step 3.2.2.1

Factor $2$ out of $2 x$ .

$8 x^{2} + 2 (x) - 1 = 0$

Step 3.2.2.2

Rewrite $2$ as $- 2$ plus $4$

$8 x^{2} + (- 2 + 4) x - 1 = 0$

Step 3.2.2.3

Apply the distributive property.

$8 x^{2} - 2 x + 4 x - 1 = 0$

Step 3.2.3

Factor out the greatest common factor from each group.

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Step 3.2.3.1

Group the first two terms and the last two terms.

$(8 x^{2} - 2 x) + 4 x - 1 = 0$

Step 3.2.3.2

Factor out the greatest common factor (GCF) from each group.

$2 x (4 x - 1) + 1 (4 x - 1) = 0$

Step 3.2.4

Factor the polynomial by factoring out the greatest common factor, $4 x - 1$ .

$(4 x - 1) (2 x + 1) = 0$

Step 3.3

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

$4 x - 1 = 0$

$2 x + 1 = 0$

Step 3.4

Set $4 x - 1$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $4 x - 1$ equal to $0$ .

$4 x - 1 = 0$

Step 3.4.2

Solve $4 x - 1 = 0$ for $x$ .

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Step 3.4.2.1

Add $1$ to both sides of the equation.

$4 x = 1$

Step 3.4.2.2

Divide each term in $4 x = 1$ by $4$ and simplify.

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Step 3.4.2.2.1

Divide each term in $4 x = 1$ by $4$ .

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

Step 3.4.2.2.2

Simplify the left side.

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Step 3.4.2.2.2.1

Cancel the common factor of $4$ .

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Step 3.4.2.2.2.1.1

Cancel the common factor.

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

Step 3.4.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.5

Set $2 x + 1$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 3.5.2

Solve $2 x + 1 = 0$ for $x$ .

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Step 3.5.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 3.5.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 3.5.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

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

Step 3.5.2.2.2

Simplify the left side.

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Step 3.5.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.5.2.2.2.1.1

Cancel the common factor.

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

Step 3.5.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.5.2.2.3

Simplify the right side.

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Step 3.5.2.2.3.1

Move the negative in front of the fraction.

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

Step 3.6

The final solution is all the values that make $(4 x - 1) (2 x + 1) = 0$ true.

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.25, - 0.5$