Finite Math Examples

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

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

Step 1

Rewrite the equation as

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

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

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

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

Step 2

Factor

0.002

$0.002$ out of

0.002 x^{2} + 2 x + 5000

$0.002 x^{2} + 2 x + 5000$ .

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Step 2.1

Factor

0.002

$0.002$ out of

0.002 x^{2}

$0.002 x^{2}$ .

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

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

Step 2.2

Factor

0.002

$0.002$ out of

2 x

$2 x$ .

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

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

Step 2.3

Factor

0.002

$0.002$ out of

5000

$5000$ .

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

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

Step 2.4

Factor

0.002

$0.002$ out of

0.002 x^{2} + 0.002 (1000 x)

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

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

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

Step 2.5

Factor

0.002

$0.002$ out of

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

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

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

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

y = 0.002 (x^{2} + 1000 x + 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

$x^{2} + 1000 x + 2500000 = 0$

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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}

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

Step 3.1.2

Substitute the values

a = 1

$a = 1$ ,

b = 1000

$b = 1000$ , and

c = 2500000

$c = 2500000$ into the quadratic formula and solve for

x

$x$ .

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

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

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

$1000$ to the power of

2

$2$ .

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

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

Step 3.1.3.1.2

Multiply

- 4 \cdot 1 \cdot 2500000

$- 4 \cdot 1 \cdot 2500000$ .

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Step 3.1.3.1.2.1

Multiply

- 4

$- 4$ by

1

$1$ .

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

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

Step 3.1.3.1.2.2

Multiply

- 4

$- 4$ by

2500000

$2500000$ .

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

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

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

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

Step 3.1.3.1.3

Subtract

10000000

$10000000$ from

1000000

$1000000$ .

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

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

Step 3.1.3.1.4

Rewrite

- 9000000

$- 9000000$ as

- 1 (9000000)

$- 1 (9000000)$ .

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

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

Step 3.1.3.1.5

Rewrite

\sqrt{- 1 (9000000)}

$\sqrt{- 1 (9000000)}$ as

\sqrt{- 1} \cdot \sqrt{9000000}

$\sqrt{- 1} \cdot \sqrt{9000000}$ .

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

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

Step 3.1.3.1.6

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

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

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

Step 3.1.3.1.7

Rewrite

9000000

$9000000$ as

3000^{2}

$3000^{2}$ .

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

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

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}

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

Step 3.1.3.1.9

Move

3000

$3000$ to the left of

i

$i$ .

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

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

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

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

Step 3.1.3.2

Multiply

2

$2$ by

1

$1$ .

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

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

Step 3.1.3.3

Simplify

\frac{- 1000 \pm 3000 i}{2}

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

x = - 500 \pm 1500 i

$x = - 500 \pm 1500 i$

x = - 500 \pm 1500 i

$x = - 500 \pm 1500 i$

x = - 500 \pm 1500 i

$x = - 500 \pm 1500 i$

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))

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

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

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