Finite Math Examples

$0.002 x^{2} + 2 x + 5000 = y$

Step 1

Rewrite the equation as $y = 0.002 x^{2} + 2 x + 5000$ .

$y = 0.002 x^{2} + 2 x + 5000$

Step 2

Factor $0.002$ out of $0.002 x^{2} + 2 x + 5000$ .

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Step 2.1

Factor $0.002$ out of $0.002 x^{2}$ .

$y = 0.002 (x^{2}) + 2 x + 5000$

Step 2.2

Factor $0.002$ out of $2 x$ .

$y = 0.002 (x^{2}) + 0.002 (1000 x) + 5000$

Step 2.3

Factor $0.002$ out of $5000$ .

$y = 0.002 x^{2} + 0.002 (1000 x) + 0.002 \cdot 2500000$

Step 2.4

Factor $0.002$ out of $0.002 x^{2} + 0.002 (1000 x)$ .

$y = 0.002 (x^{2} + 1000 x) + 0.002 \cdot 2500000$

Step 2.5

Factor $0.002$ out of $0.002 (x^{2} + 1000 x) + 0.002 \cdot 2500000$ .

$y = 0.002 (x^{2} + 1000 x + 2500000)$

Step 3

Factor over the complex numbers.

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Step 3.1

Use the quadratic formula to find the roots for $x^{2} + 1000 x + 2500000 = 0$

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Step 3.1.1

Use the quadratic formula to find the solutions.

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

Step 3.1.2

Substitute the values $a = 1$ , $b = 1000$ , and $c = 2500000$ into the quadratic formula and solve for $x$ .

$y = \frac{- 1000 \pm \sqrt{1000^{2} - 4 \cdot (1 \cdot 2500000)}}{2 \cdot 1}$

Step 3.1.3

Simplify.

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Step 3.1.3.1

Simplify the numerator.

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Step 3.1.3.1.1

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

$x = \frac{- 1000 \pm \sqrt{1000000 - 4 \cdot 1 \cdot 2500000}}{2 \cdot 1}$

Step 3.1.3.1.2

Multiply $- 4 \cdot 1 \cdot 2500000$ .

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Step 3.1.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1000 \pm \sqrt{1000000 - 4 \cdot 2500000}}{2 \cdot 1}$

Step 3.1.3.1.2.2

Multiply $- 4$ by $2500000$ .

$x = \frac{- 1000 \pm \sqrt{1000000 - 10000000}}{2 \cdot 1}$

Step 3.1.3.1.3

Subtract $10000000$ from $1000000$ .

$x = \frac{- 1000 \pm \sqrt{- 9000000}}{2 \cdot 1}$

Step 3.1.3.1.4

Rewrite $- 9000000$ as $- 1 (9000000)$ .

$x = \frac{- 1000 \pm \sqrt{- 1 \cdot 9000000}}{2 \cdot 1}$

Step 3.1.3.1.5

Rewrite $\sqrt{- 1 (9000000)}$ as $\sqrt{- 1} \cdot \sqrt{9000000}$ .

$x = \frac{- 1000 \pm \sqrt{- 1} \cdot \sqrt{9000000}}{2 \cdot 1}$

Step 3.1.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1000 \pm i \cdot \sqrt{9000000}}{2 \cdot 1}$

Step 3.1.3.1.7

Rewrite $9000000$ as $3000^{2}$ .

$x = \frac{- 1000 \pm i \cdot \sqrt{3000^{2}}}{2 \cdot 1}$

Step 3.1.3.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 1000 \pm i \cdot 3000}{2 \cdot 1}$

Step 3.1.3.1.9

Move $3000$ to the left of $i$ .

$x = \frac{- 1000 \pm 3000 i}{2 \cdot 1}$

Step 3.1.3.2

Multiply $2$ by $1$ .

$x = \frac{- 1000 \pm 3000 i}{2}$

Step 3.1.3.3

Simplify $\frac{- 1000 \pm 3000 i}{2}$ .

$x = - 500 \pm 1500 i$

Step 3.2

Find the factors from the roots, then multiply the factors together.

$y = 0.002 (x - (- 500 + 1500 i)) (x - (- 500 - 1500 i))$