Calculus Examples

$c = 8 x + \frac{100000}{x}$

Step 1

Rewrite the equation as $8 x + \frac{100000}{x} = c$ .

$8 x + \frac{100000}{x} = c$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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 2.2

The LCM of one and any expression is the expression.

$x$

Step 3

Multiply each term in $8 x + \frac{100000}{x} = c$ by $x$ to eliminate the fractions.

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Step 3.1

Multiply each term in $8 x + \frac{100000}{x} = c$ by $x$ .

$8 x \cdot x + \frac{100000}{x} x = c x$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

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Step 3.2.1.1.1

Move $x$ .

$8 (x \cdot x) + \frac{100000}{x} x = c x$

Step 3.2.1.1.2

Multiply $x$ by $x$ .

$8 x^{2} + \frac{100000}{x} x = c x$

Step 3.2.1.2

Cancel the common factor of $x$ .

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Step 3.2.1.2.1

Cancel the common factor.

$8 x^{2} + \frac{100000}{x} x = c x$

Step 3.2.1.2.2

Rewrite the expression.

$8 x^{2} + 100000 = c x$

Step 4

Solve the equation.

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Step 4.1

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

$8 x^{2} + 100000 - c x = 0$

Step 4.2

Use the quadratic formula to find the solutions.

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

Step 4.3

Substitute the values $a = 8$ , $b = - c$ , and $c = 100000$ into the quadratic formula and solve for $x$ .

$\frac{c \pm \sqrt{{(- c)}^{2} - 4 \cdot (8 \cdot 100000)}}{2 \cdot 8}$

Step 4.4

Simplify.

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Step 4.4.1

Simplify the numerator.

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Step 4.4.1.1

Apply the product rule to $- c$ .

$x = \frac{c \pm \sqrt{{(- 1)}^{2} c^{2} - 4 \cdot 8 \cdot 100000}}{2 \cdot 8}$

Step 4.4.1.2

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

$x = \frac{c \pm \sqrt{1 c^{2} - 4 \cdot 8 \cdot 100000}}{2 \cdot 8}$

Step 4.4.1.3

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

$x = \frac{c \pm \sqrt{c^{2} - 4 \cdot 8 \cdot 100000}}{2 \cdot 8}$

Step 4.4.1.4

Multiply $- 4 \cdot 8 \cdot 100000$ .

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Step 4.4.1.4.1

Multiply $- 4$ by $8$ .

$x = \frac{c \pm \sqrt{c^{2} - 32 \cdot 100000}}{2 \cdot 8}$

Step 4.4.1.4.2

Multiply $- 32$ by $100000$ .

$x = \frac{c \pm \sqrt{c^{2} - 3200000}}{2 \cdot 8}$

Step 4.4.2

Multiply $2$ by $8$ .

$x = \frac{c \pm \sqrt{c^{2} - 3200000}}{16}$

Step 4.5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Apply the product rule to $- c$ .

$x = \frac{c \pm \sqrt{{(- 1)}^{2} c^{2} - 4 \cdot 8 \cdot 100000}}{2 \cdot 8}$

Step 4.5.1.2

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

$x = \frac{c \pm \sqrt{1 c^{2} - 4 \cdot 8 \cdot 100000}}{2 \cdot 8}$

Step 4.5.1.3

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

$x = \frac{c \pm \sqrt{c^{2} - 4 \cdot 8 \cdot 100000}}{2 \cdot 8}$

Step 4.5.1.4

Multiply $- 4 \cdot 8 \cdot 100000$ .

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Step 4.5.1.4.1

Multiply $- 4$ by $8$ .

$x = \frac{c \pm \sqrt{c^{2} - 32 \cdot 100000}}{2 \cdot 8}$

Step 4.5.1.4.2

Multiply $- 32$ by $100000$ .

$x = \frac{c \pm \sqrt{c^{2} - 3200000}}{2 \cdot 8}$

Step 4.5.2

Multiply $2$ by $8$ .

$x = \frac{c \pm \sqrt{c^{2} - 3200000}}{16}$

Step 4.5.3

Change the $\pm$ to $+$ .

$x = \frac{c + \sqrt{c^{2} - 3200000}}{16}$

Step 4.6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 4.6.1

Simplify the numerator.

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Step 4.6.1.1

Apply the product rule to $- c$ .

$x = \frac{c \pm \sqrt{{(- 1)}^{2} c^{2} - 4 \cdot 8 \cdot 100000}}{2 \cdot 8}$

Step 4.6.1.2

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

$x = \frac{c \pm \sqrt{1 c^{2} - 4 \cdot 8 \cdot 100000}}{2 \cdot 8}$

Step 4.6.1.3

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

$x = \frac{c \pm \sqrt{c^{2} - 4 \cdot 8 \cdot 100000}}{2 \cdot 8}$

Step 4.6.1.4

Multiply $- 4 \cdot 8 \cdot 100000$ .

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Step 4.6.1.4.1

Multiply $- 4$ by $8$ .

$x = \frac{c \pm \sqrt{c^{2} - 32 \cdot 100000}}{2 \cdot 8}$

Step 4.6.1.4.2

Multiply $- 32$ by $100000$ .

$x = \frac{c \pm \sqrt{c^{2} - 3200000}}{2 \cdot 8}$

Step 4.6.2

Multiply $2$ by $8$ .

$x = \frac{c \pm \sqrt{c^{2} - 3200000}}{16}$

Step 4.6.3

Change the $\pm$ to $-$ .

$x = \frac{c - \sqrt{c^{2} - 3200000}}{16}$

Step 4.7

The final answer is the combination of both solutions.

$x = \frac{c + \sqrt{c^{2} - 3200000}}{16}$

$x = \frac{c - \sqrt{c^{2} - 3200000}}{16}$

$x = \frac{c + \sqrt{c^{2} - 3200000}}{16}$

$x = \frac{c - \sqrt{c^{2} - 3200000}}{16}$

Step 5

To rewrite as a function of $c$ , write the equation so that $x$ is by itself on one side of the equal sign and an expression involving only $c$ is on the other side.

$f (c) = \frac{c + \sqrt{c^{2} - 3200000}}{16}, \frac{c - \sqrt{c^{2} - 3200000}}{16}$