Basic Math Examples

$p = - 4 p^{2} + 8000$

Step 1

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

$p + 4 p^{2} = 8000$

Step 2

Subtract $8000$ from both sides of the equation.

$p + 4 p^{2} - 8000 = 0$

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 $a = 4$ , $b = 1$ , and $c = - 8000$ into the quadratic formula and solve for $p$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (4 \cdot - 8000)}}{2 \cdot 4}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

One to any power is one.

$p = \frac{- 1 \pm \sqrt{1 - 4 \cdot 4 \cdot - 8000}}{2 \cdot 4}$

Step 5.1.2

Multiply $- 4 \cdot 4 \cdot - 8000$ .

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Step 5.1.2.1

Multiply $- 4$ by $4$ .

$p = \frac{- 1 \pm \sqrt{1 - 16 \cdot - 8000}}{2 \cdot 4}$

Step 5.1.2.2

Multiply $- 16$ by $- 8000$ .

$p = \frac{- 1 \pm \sqrt{1 + 128000}}{2 \cdot 4}$

Step 5.1.3

Add $1$ and $128000$ .

$p = \frac{- 1 \pm \sqrt{128001}}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$p = \frac{- 1 \pm \sqrt{128001}}{8}$

Step 6

The final answer is the combination of both solutions.

$p = - \frac{1 - \sqrt{128001}}{8}, - \frac{1 + \sqrt{128001}}{8}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$p = - \frac{1 - \sqrt{128001}}{8}, - \frac{1 + \sqrt{128001}}{8}$

Decimal Form:

$p = 44.59653424 \dots, - 44.84653424 \dots$