Pre-Algebra Examples

$0.06 x (12720 - x) = 12$

Step 1

Divide each term in $0.06 x (12720 - x) = 12$ by $0.06$ and simplify.

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Step 1.1

Divide each term in $0.06 x (12720 - x) = 12$ by $0.06$ .

$\frac{0.06 x (12720 - x)}{0.06} = \frac{12}{0.06}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify terms.

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Step 1.2.1.1

Cancel the common factor of $0.06$ .

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Step 1.2.1.1.1

Cancel the common factor.

$\frac{0.06 x (12720 - x)}{0.06} = \frac{12}{0.06}$

Step 1.2.1.1.2

Divide $x (12720 - x)$ by $1$ .

$x (12720 - x) = \frac{12}{0.06}$

Step 1.2.1.2

Apply the distributive property.

$x \cdot 12720 + x (- x) = \frac{12}{0.06}$

Step 1.2.1.3

Reorder.

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Step 1.2.1.3.1

Move $12720$ to the left of $x$ .

$12720 \cdot x + x (- x) = \frac{12}{0.06}$

Step 1.2.1.3.2

Rewrite using the commutative property of multiplication.

$12720 \cdot x - x \cdot x = \frac{12}{0.06}$

Step 1.2.2

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

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Step 1.2.2.1

Move $x$ .

$12720 x - (x \cdot x) = \frac{12}{0.06}$

Step 1.2.2.2

Multiply $x$ by $x$ .

$12720 x - x^{2} = \frac{12}{0.06}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $12$ by $0.06$ .

$12720 x - x^{2} = 200$

Step 2

Subtract $200$ from both sides of the equation.

$12720 x - x^{2} - 200 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = - 1$ , $b = 12720$ , and $c = - 200$ into the quadratic formula and solve for $x$ .

$\frac{- 12720 \pm \sqrt{12720^{2} - 4 \cdot (- 1 \cdot - 200)}}{2 \cdot - 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $12720$ to the power of $2$ .

$x = \frac{- 12720 \pm \sqrt{161798400 - 4 \cdot - 1 \cdot - 200}}{2 \cdot - 1}$

Step 5.1.2

Multiply $- 4 \cdot - 1 \cdot - 200$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 12720 \pm \sqrt{161798400 + 4 \cdot - 200}}{2 \cdot - 1}$

Step 5.1.2.2

Multiply $4$ by $- 200$ .

$x = \frac{- 12720 \pm \sqrt{161798400 - 800}}{2 \cdot - 1}$

Step 5.1.3

Subtract $800$ from $161798400$ .

$x = \frac{- 12720 \pm \sqrt{161797600}}{2 \cdot - 1}$

Step 5.1.4

Rewrite $161797600$ as $20^{2} \cdot 404494$ .

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Step 5.1.4.1

Factor $400$ out of $161797600$ .

$x = \frac{- 12720 \pm \sqrt{400 (404494)}}{2 \cdot - 1}$

Step 5.1.4.2

Rewrite $400$ as $20^{2}$ .

$x = \frac{- 12720 \pm \sqrt{20^{2} \cdot 404494}}{2 \cdot - 1}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{- 12720 \pm 20 \sqrt{404494}}{2 \cdot - 1}$

Step 5.2

Multiply $2$ by $- 1$ .

$x = \frac{- 12720 \pm 20 \sqrt{404494}}{- 2}$

Step 5.3

Simplify $\frac{- 12720 \pm 20 \sqrt{404494}}{- 2}$ .

$x = 6360 \pm 10 \sqrt{404494}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $12720$ to the power of $2$ .

$x = \frac{- 12720 \pm \sqrt{161798400 - 4 \cdot - 1 \cdot - 200}}{2 \cdot - 1}$

Step 6.1.2

Multiply $- 4 \cdot - 1 \cdot - 200$ .

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Step 6.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 12720 \pm \sqrt{161798400 + 4 \cdot - 200}}{2 \cdot - 1}$

Step 6.1.2.2

Multiply $4$ by $- 200$ .

$x = \frac{- 12720 \pm \sqrt{161798400 - 800}}{2 \cdot - 1}$

Step 6.1.3

Subtract $800$ from $161798400$ .

$x = \frac{- 12720 \pm \sqrt{161797600}}{2 \cdot - 1}$

Step 6.1.4

Rewrite $161797600$ as $20^{2} \cdot 404494$ .

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Step 6.1.4.1

Factor $400$ out of $161797600$ .

$x = \frac{- 12720 \pm \sqrt{400 (404494)}}{2 \cdot - 1}$

Step 6.1.4.2

Rewrite $400$ as $20^{2}$ .

$x = \frac{- 12720 \pm \sqrt{20^{2} \cdot 404494}}{2 \cdot - 1}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{- 12720 \pm 20 \sqrt{404494}}{2 \cdot - 1}$

Step 6.2

Multiply $2$ by $- 1$ .

$x = \frac{- 12720 \pm 20 \sqrt{404494}}{- 2}$

Step 6.3

Simplify $\frac{- 12720 \pm 20 \sqrt{404494}}{- 2}$ .

$x = 6360 \pm 10 \sqrt{404494}$

Step 6.4

Change the $\pm$ to $+$ .

$x = 6360 + 10 \sqrt{404494}$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $12720$ to the power of $2$ .

$x = \frac{- 12720 \pm \sqrt{161798400 - 4 \cdot - 1 \cdot - 200}}{2 \cdot - 1}$

Step 7.1.2

Multiply $- 4 \cdot - 1 \cdot - 200$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 12720 \pm \sqrt{161798400 + 4 \cdot - 200}}{2 \cdot - 1}$

Step 7.1.2.2

Multiply $4$ by $- 200$ .

$x = \frac{- 12720 \pm \sqrt{161798400 - 800}}{2 \cdot - 1}$

Step 7.1.3

Subtract $800$ from $161798400$ .

$x = \frac{- 12720 \pm \sqrt{161797600}}{2 \cdot - 1}$

Step 7.1.4

Rewrite $161797600$ as $20^{2} \cdot 404494$ .

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Step 7.1.4.1

Factor $400$ out of $161797600$ .

$x = \frac{- 12720 \pm \sqrt{400 (404494)}}{2 \cdot - 1}$

Step 7.1.4.2

Rewrite $400$ as $20^{2}$ .

$x = \frac{- 12720 \pm \sqrt{20^{2} \cdot 404494}}{2 \cdot - 1}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{- 12720 \pm 20 \sqrt{404494}}{2 \cdot - 1}$

Step 7.2

Multiply $2$ by $- 1$ .

$x = \frac{- 12720 \pm 20 \sqrt{404494}}{- 2}$

Step 7.3

Simplify $\frac{- 12720 \pm 20 \sqrt{404494}}{- 2}$ .

$x = 6360 \pm 10 \sqrt{404494}$

Step 7.4

Change the $\pm$ to $-$ .

$x = 6360 - 10 \sqrt{404494}$

Step 8

The final answer is the combination of both solutions.

$x = 6360 + 10 \sqrt{404494}, 6360 - 10 \sqrt{404494}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = 6360 + 10 \sqrt{404494}, 6360 - 10 \sqrt{404494}$

Decimal Form:

$x = 12719.98427671 \dots, 0.01572328 \dots$