Finite Math Examples

$r = 1000 x - 200 x^{2}$

Step 1

Rewrite the equation as $1000 x - 200 x^{2} = r$ .

$1000 x - 200 x^{2} = r$

Step 2

Subtract $r$ from both sides of the equation.

$1000 x - 200 x^{2} - r = 0$

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 $a = - 200$ , $b = 1000$ , and $c = - r$ into the quadratic formula and solve for $x$ .

$\frac{- 1000 \pm \sqrt{1000^{2} - 4 \cdot (- 200 \cdot (- r))}}{2 \cdot - 200}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $1000$ to the power of $2$ .

$x = \frac{- 1000 \pm \sqrt{1000000 - 4 \cdot - 200 \cdot (- 1 r)}}{2 \cdot - 200}$

Step 5.1.2

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

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Step 5.1.2.1

Multiply $- 4$ by $- 200$ .

$x = \frac{- 1000 \pm \sqrt{1000000 + 800 \cdot (- 1 r)}}{2 \cdot - 200}$

Step 5.1.2.2

Multiply $800$ by $- 1$ .

$x = \frac{- 1000 \pm \sqrt{1000000 - 800 r}}{2 \cdot - 200}$

Step 5.1.3

Factor $800$ out of $1000000 - 800 r$ .

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Step 5.1.3.1

Factor $800$ out of $1000000$ .

$x = \frac{- 1000 \pm \sqrt{800 (1250) - 800 r}}{2 \cdot - 200}$

Step 5.1.3.2

Factor $800$ out of $- 800 r$ .

$x = \frac{- 1000 \pm \sqrt{800 (1250) + 800 (- r)}}{2 \cdot - 200}$

Step 5.1.3.3

Factor $800$ out of $800 (1250) + 800 (- r)$ .

$x = \frac{- 1000 \pm \sqrt{800 (1250 - r)}}{2 \cdot - 200}$

Step 5.1.4

Rewrite $800 (1250 - r)$ as $20^{2} \cdot (2 (1250 - r))$ .

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Step 5.1.4.1

Factor $400$ out of $800$ .

$x = \frac{- 1000 \pm \sqrt{400 (2) (1250 - r)}}{2 \cdot - 200}$

Step 5.1.4.2

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

$x = \frac{- 1000 \pm \sqrt{20^{2} \cdot (2 (1250 - r))}}{2 \cdot - 200}$

Step 5.1.4.3

Add parentheses.

$x = \frac{- 1000 \pm \sqrt{20^{2} \cdot (2 (1250 - r))}}{2 \cdot - 200}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{- 1000 \pm 20 \sqrt{2 (1250 - r)}}{2 \cdot - 200}$

Step 5.2

Multiply $2$ by $- 200$ .

$x = \frac{- 1000 \pm 20 \sqrt{2 (1250 - r)}}{- 400}$

Step 5.3

Simplify $\frac{- 1000 \pm 20 \sqrt{2 (1250 - r)}}{- 400}$ .

$x = \frac{50 \pm \sqrt{2 (1250 - r)}}{20}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{50 + \sqrt{2 (1250 - r)}}{20}$

$x = \frac{50 - \sqrt{2 (1250 - r)}}{20}$