Pre-Algebra Examples

$c - \frac{1}{4 c} = 49.95$

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.

$1, 4 c, 1$

Step 1.2

The LCM of one and any expression is the expression.

$4 c$

Step 2

Multiply each term in $c - \frac{1}{4 c} = 49.95$ by $4 c$ to eliminate the fractions.

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Step 2.1

Multiply each term in $c - \frac{1}{4 c} = 49.95$ by $4 c$ .

$c (4 c) - \frac{1}{4 c} (4 c) = 49.95 (4 c)$

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

Rewrite using the commutative property of multiplication.

$4 c \cdot c - \frac{1}{4 c} (4 c) = 49.95 (4 c)$

Step 2.2.1.2

Multiply $c$ by $c$ by adding the exponents.

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Step 2.2.1.2.1

Move $c$ .

$4 (c \cdot c) - \frac{1}{4 c} (4 c) = 49.95 (4 c)$

Step 2.2.1.2.2

Multiply $c$ by $c$ .

$4 c^{2} - \frac{1}{4 c} (4 c) = 49.95 (4 c)$

Step 2.2.1.3

Cancel the common factor of $4 c$ .

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Step 2.2.1.3.1

Move the leading negative in $- \frac{1}{4 c}$ into the numerator.

$4 c^{2} + \frac{- 1}{4 c} (4 c) = 49.95 (4 c)$

Step 2.2.1.3.2

Cancel the common factor.

$4 c^{2} + \frac{- 1}{4 c} (4 c) = 49.95 (4 c)$

Step 2.2.1.3.3

Rewrite the expression.

$4 c^{2} - 1 = 49.95 (4 c)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply $4$ by $49.95$ .

$4 c^{2} - 1 = 199.8 c$

Step 3

Solve the equation.

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Step 3.1

Subtract $199.8 c$ from both sides of the equation.

$4 c^{2} - 1 - 199.8 c = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Reorder terms.

$4 c^{2} - 199.8 c - 1 = 0$

Step 3.2.2

Factor $0.2$ out of $4 c^{2} - 199.8 c - 1$ .

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Step 3.2.2.1

Factor $0.2$ out of $4 c^{2}$ .

$0.2 (20 c^{2}) - 199.8 c - 1 = 0$

Step 3.2.2.2

Factor $0.2$ out of $- 199.8 c$ .

$0.2 (20 c^{2}) + 0.2 (- 999 c) - 1 = 0$

Step 3.2.2.3

Factor $0.2$ out of $- 1$ .

$0.2 (20 c^{2}) + 0.2 (- 999 c) + 0.2 (- 5) = 0$

Step 3.2.2.4

Factor $0.2$ out of $0.2 (20 c^{2}) + 0.2 (- 999 c)$ .

$0.2 (20 c^{2} - 999 c) + 0.2 (- 5) = 0$

Step 3.2.2.5

Factor $0.2$ out of $0.2 (20 c^{2} - 999 c) + 0.2 (- 5)$ .

$0.2 (20 c^{2} - 999 c - 5) = 0$

Step 3.3

Divide each term in $0.2 (20 c^{2} - 999 c - 5) = 0$ by $0.2$ and simplify.

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Step 3.3.1

Divide each term in $0.2 (20 c^{2} - 999 c - 5) = 0$ by $0.2$ .

$\frac{0.2 (20 c^{2} - 999 c - 5)}{0.2} = \frac{0}{0.2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $0.2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{0.2 (20 c^{2} - 999 c - 5)}{0.2} = \frac{0}{0.2}$

Step 3.3.2.1.2

Divide $20 c^{2} - 999 c - 5$ by $1$ .

$20 c^{2} - 999 c - 5 = \frac{0}{0.2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $0$ by $0.2$ .

$20 c^{2} - 999 c - 5 = 0$

Step 3.4

Use the quadratic formula to find the solutions.

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

Step 3.5

Substitute the values $a = 20$ , $b = - 999$ , and $c = - 5$ into the quadratic formula and solve for $c$ .

$\frac{999 \pm \sqrt{{(- 999)}^{2} - 4 \cdot (20 \cdot - 5)}}{2 \cdot 20}$

Step 3.6

Simplify.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Raise $- 999$ to the power of $2$ .

$c = \frac{999 \pm \sqrt{998001 - 4 \cdot 20 \cdot - 5}}{2 \cdot 20}$

Step 3.6.1.2

Multiply $- 4 \cdot 20 \cdot - 5$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $20$ .

$c = \frac{999 \pm \sqrt{998001 - 80 \cdot - 5}}{2 \cdot 20}$

Step 3.6.1.2.2

Multiply $- 80$ by $- 5$ .

$c = \frac{999 \pm \sqrt{998001 + 400}}{2 \cdot 20}$

Step 3.6.1.3

Add $998001$ and $400$ .

$c = \frac{999 \pm \sqrt{998401}}{2 \cdot 20}$

Step 3.6.2

Multiply $2$ by $20$ .

$c = \frac{999 \pm \sqrt{998401}}{40}$

Step 3.7

The final answer is the combination of both solutions.

$c = \frac{999 + \sqrt{998401}}{40}, \frac{999 - \sqrt{998401}}{40}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$c = \frac{999 + \sqrt{998401}}{40}, \frac{999 - \sqrt{998401}}{40}$

Decimal Form:

$c = 49.95500450 \dots, - 0.00500450 \dots$