Precalculus Examples

$c (x) = \frac{1500}{x} + 105 + x$

Step 1

Set $\frac{1500}{x} + 105 + x$ equal to $0$ .

$\frac{1500}{x} + 105 + x = 0$

Step 2

Solve for $x$ .

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Step 2.1

Find the LCD of the terms in the equation.

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Step 2.1.1

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

$x, 1, 1, 1$

Step 2.1.2

The LCM of one and any expression is the expression.

$x$

Step 2.2

Multiply each term in $\frac{1500}{x} + 105 + x = 0$ by $x$ to eliminate the fractions.

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Step 2.2.1

Multiply each term in $\frac{1500}{x} + 105 + x = 0$ by $x$ .

$\frac{1500}{x} x + 105 x + x \cdot x = 0 x$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Cancel the common factor of $x$ .

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Step 2.2.2.1.1.1

Cancel the common factor.

$\frac{1500}{x} x + 105 x + x \cdot x = 0 x$

Step 2.2.2.1.1.2

Rewrite the expression.

$1500 + 105 x + x \cdot x = 0 x$

Step 2.2.2.1.2

Multiply $x$ by $x$ .

$1500 + 105 x + x^{2} = 0 x$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Multiply $0$ by $x$ .

$1500 + 105 x + x^{2} = 0$

Step 2.3

Solve the equation.

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Step 2.3.1

Use the quadratic formula to find the solutions.

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

Step 2.3.2

Substitute the values $a = 1$ , $b = 105$ , and $c = 1500$ into the quadratic formula and solve for $x$ .

$\frac{- 105 \pm \sqrt{105^{2} - 4 \cdot (1 \cdot 1500)}}{2 \cdot 1}$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Simplify the numerator.

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Step 2.3.3.1.1

Raise $105$ to the power of $2$ .

$x = \frac{- 105 \pm \sqrt{11025 - 4 \cdot 1 \cdot 1500}}{2 \cdot 1}$

Step 2.3.3.1.2

Multiply $- 4 \cdot 1 \cdot 1500$ .

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Step 2.3.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 105 \pm \sqrt{11025 - 4 \cdot 1500}}{2 \cdot 1}$

Step 2.3.3.1.2.2

Multiply $- 4$ by $1500$ .

$x = \frac{- 105 \pm \sqrt{11025 - 6000}}{2 \cdot 1}$

Step 2.3.3.1.3

Subtract $6000$ from $11025$ .

$x = \frac{- 105 \pm \sqrt{5025}}{2 \cdot 1}$

Step 2.3.3.1.4

Rewrite $5025$ as $5^{2} \cdot 201$ .

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Step 2.3.3.1.4.1

Factor $25$ out of $5025$ .

$x = \frac{- 105 \pm \sqrt{25 (201)}}{2 \cdot 1}$

Step 2.3.3.1.4.2

Rewrite $25$ as $5^{2}$ .

$x = \frac{- 105 \pm \sqrt{5^{2} \cdot 201}}{2 \cdot 1}$

Step 2.3.3.1.5

Pull terms out from under the radical.

$x = \frac{- 105 \pm 5 \sqrt{201}}{2 \cdot 1}$

Step 2.3.3.2

Multiply $2$ by $1$ .

$x = \frac{- 105 \pm 5 \sqrt{201}}{2}$

Step 2.3.4

The final answer is the combination of both solutions.

$x = - \frac{105 - 5 \sqrt{201}}{2}, - \frac{105 + 5 \sqrt{201}}{2}$

$x = \frac{- 105 \pm 5 \sqrt{201}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{- 105 \pm 5 \sqrt{201}}{2}$

Decimal Form:

$x = - 17.05638280 \dots, - 87.94361719 \dots$

Step 4