Finite Math Examples

$Q = - 4 Q^{2} + 200 Q + 20000$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Add $4 Q^{2}$ to both sides of the equation.

$Q + 4 Q^{2} = 200 Q + 20000$

Step 1.2

Subtract $200 Q$ from both sides of the equation.

$Q + 4 Q^{2} - 200 Q = 20000$

Step 1.3

Subtract $20000$ from both sides of the equation.

$Q + 4 Q^{2} - 200 Q - 20000 = 0$

Step 2

Subtract $200 Q$ from $Q$ .

$4 Q^{2} - 199 Q - 20000 = 0$

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 $a = 4$ , $b = - 199$ , and $c = - 20000$ into the quadratic formula and solve for $Q$ .

$\frac{199 \pm \sqrt{{(- 199)}^{2} - 4 \cdot (4 \cdot - 20000)}}{2 \cdot 4}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 199$ to the power of $2$ .

$Q = \frac{199 \pm \sqrt{39601 - 4 \cdot 4 \cdot - 20000}}{2 \cdot 4}$

Step 5.1.2

Multiply $- 4 \cdot 4 \cdot - 20000$ .

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Step 5.1.2.1

Multiply $- 4$ by $4$ .

$Q = \frac{199 \pm \sqrt{39601 - 16 \cdot - 20000}}{2 \cdot 4}$

Step 5.1.2.2

Multiply $- 16$ by $- 20000$ .

$Q = \frac{199 \pm \sqrt{39601 + 320000}}{2 \cdot 4}$

Step 5.1.3

Add $39601$ and $320000$ .

$Q = \frac{199 \pm \sqrt{359601}}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$Q = \frac{199 \pm \sqrt{359601}}{8}$

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 $- 199$ to the power of $2$ .

$Q = \frac{199 \pm \sqrt{39601 - 4 \cdot 4 \cdot - 20000}}{2 \cdot 4}$

Step 6.1.2

Multiply $- 4 \cdot 4 \cdot - 20000$ .

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Step 6.1.2.1

Multiply $- 4$ by $4$ .

$Q = \frac{199 \pm \sqrt{39601 - 16 \cdot - 20000}}{2 \cdot 4}$

Step 6.1.2.2

Multiply $- 16$ by $- 20000$ .

$Q = \frac{199 \pm \sqrt{39601 + 320000}}{2 \cdot 4}$

Step 6.1.3

Add $39601$ and $320000$ .

$Q = \frac{199 \pm \sqrt{359601}}{2 \cdot 4}$

Step 6.2

Multiply $2$ by $4$ .

$Q = \frac{199 \pm \sqrt{359601}}{8}$

Step 6.3

Change the $\pm$ to $+$ .

$Q = \frac{199 + \sqrt{359601}}{8}$

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 $- 199$ to the power of $2$ .

$Q = \frac{199 \pm \sqrt{39601 - 4 \cdot 4 \cdot - 20000}}{2 \cdot 4}$

Step 7.1.2

Multiply $- 4 \cdot 4 \cdot - 20000$ .

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Step 7.1.2.1

Multiply $- 4$ by $4$ .

$Q = \frac{199 \pm \sqrt{39601 - 16 \cdot - 20000}}{2 \cdot 4}$

Step 7.1.2.2

Multiply $- 16$ by $- 20000$ .

$Q = \frac{199 \pm \sqrt{39601 + 320000}}{2 \cdot 4}$

Step 7.1.3

Add $39601$ and $320000$ .

$Q = \frac{199 \pm \sqrt{359601}}{2 \cdot 4}$

Step 7.2

Multiply $2$ by $4$ .

$Q = \frac{199 \pm \sqrt{359601}}{8}$

Step 7.3

Change the $\pm$ to $-$ .

$Q = \frac{199 - \sqrt{359601}}{8}$

Step 8

The final answer is the combination of both solutions.

$Q = \frac{199 + \sqrt{359601}}{8}, \frac{199 - \sqrt{359601}}{8}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$Q = \frac{199 + \sqrt{359601}}{8}, \frac{199 - \sqrt{359601}}{8}$

Decimal Form:

$Q = 99.83342597 \dots, - 50.08342597 \dots$