Calculus Examples

$- 6 x + \frac{24}{x} = 0$

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 $- 6 x + \frac{24}{x} = 0$ by $x$ to eliminate the fractions.

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Step 2.1

Multiply each term in $- 6 x + \frac{24}{x} = 0$ by $x$ .

$- 6 x \cdot x + \frac{24}{x} x = 0 x$

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 $x$ .

$- 6 (x \cdot x) + \frac{24}{x} x = 0 x$

Step 2.2.1.1.2

Multiply $x$ by $x$ .

$- 6 x^{2} + \frac{24}{x} x = 0 x$

Step 2.2.1.2

Cancel the common factor of $x$ .

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Step 2.2.1.2.1

Cancel the common factor.

$- 6 x^{2} + \frac{24}{x} x = 0 x$

Step 2.2.1.2.2

Rewrite the expression.

$- 6 x^{2} + 24 = 0 x$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply $0$ by $x$ .

$- 6 x^{2} + 24 = 0$

Step 3

Solve the equation.

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Step 3.1

Subtract $24$ from both sides of the equation.

$- 6 x^{2} = - 24$

Step 3.2

Divide each term in $- 6 x^{2} = - 24$ by $- 6$ and simplify.

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Step 3.2.1

Divide each term in $- 6 x^{2} = - 24$ by $- 6$ .

$\frac{- 6 x^{2}}{- 6} = \frac{- 24}{- 6}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 6$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 6 x^{2}}{- 6} = \frac{- 24}{- 6}$

Step 3.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 24}{- 6}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 24$ by $- 6$ .

$x^{2} = 4$

Step 3.3

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

$x = \pm \sqrt{4}$

Step 3.4

Simplify $\pm \sqrt{4}$ .

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Step 3.4.1

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

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

Step 3.4.2

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

$x = \pm 2$

Step 3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Step 3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

Step 3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$