Calculus Examples

$x + \frac{30}{x} = s$

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 $x + \frac{30}{x} = s$ by $x$ to eliminate the fractions.

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Step 2.1

Multiply each term in $x + \frac{30}{x} = s$ by $x$ .

$x \cdot x + \frac{30}{x} x = s x$

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 $x$ .

$x^{2} + \frac{30}{x} x = s x$

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.

$x^{2} + \frac{30}{x} x = s x$

Step 2.2.1.2.2

Rewrite the expression.

$x^{2} + 30 = s x$

Step 3

Solve the equation.

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Step 3.1

Subtract $s x$ from both sides of the equation.

$x^{2} + 30 - s x = 0$

Step 3.2

Use the quadratic formula to find the solutions.

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

Step 3.3

Substitute the values $a = 1$ , $b = - s$ , and $c = 30$ into the quadratic formula and solve for $x$ .

$\frac{s \pm \sqrt{{(- s)}^{2} - 4 \cdot (1 \cdot 30)}}{2 \cdot 1}$

Step 3.4

Simplify.

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Step 3.4.1

Simplify the numerator.

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Step 3.4.1.1

Apply the product rule to $- s$ .

$x = \frac{s \pm \sqrt{{(- 1)}^{2} s^{2} - 4 \cdot 1 \cdot 30}}{2 \cdot 1}$

Step 3.4.1.2

Raise $- 1$ to the power of $2$ .

$x = \frac{s \pm \sqrt{1 s^{2} - 4 \cdot 1 \cdot 30}}{2 \cdot 1}$

Step 3.4.1.3

Multiply $s^{2}$ by $1$ .

$x = \frac{s \pm \sqrt{s^{2} - 4 \cdot 1 \cdot 30}}{2 \cdot 1}$

Step 3.4.1.4

Multiply $- 4 \cdot 1 \cdot 30$ .

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Step 3.4.1.4.1

Multiply $- 4$ by $1$ .

$x = \frac{s \pm \sqrt{s^{2} - 4 \cdot 30}}{2 \cdot 1}$

Step 3.4.1.4.2

Multiply $- 4$ by $30$ .

$x = \frac{s \pm \sqrt{s^{2} - 120}}{2 \cdot 1}$

Step 3.4.2

Multiply $2$ by $1$ .

$x = \frac{s \pm \sqrt{s^{2} - 120}}{2}$

Step 3.5

The final answer is the combination of both solutions.

$x = \frac{s + \sqrt{s^{2} - 120}}{2}$

$x = \frac{s - \sqrt{s^{2} - 120}}{2}$

$x = \frac{s + \sqrt{s^{2} - 120}}{2}$

$x = \frac{s - \sqrt{s^{2} - 120}}{2}$