Calculus Examples

$0.012 q^{2} - 0.6 q + 60$

Step 1

Factor $0.012$ out of $0.012 q^{2} - 0.6 q + 60$ .

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Step 1.1

Factor $0.012$ out of $0.012 q^{2}$ .

$0.012 (q^{2}) - 0.6 q + 60$

Step 1.2

Factor $0.012$ out of $- 0.6 q$ .

$0.012 (q^{2}) + 0.012 (- 50 q) + 60$

Step 1.3

Factor $0.012$ out of $60$ .

$0.012 q^{2} + 0.012 (- 50 q) + 0.012 \cdot 5000$

Step 1.4

Factor $0.012$ out of $0.012 q^{2} + 0.012 (- 50 q)$ .

$0.012 (q^{2} - 50 q) + 0.012 \cdot 5000$

Step 1.5

Factor $0.012$ out of $0.012 (q^{2} - 50 q) + 0.012 \cdot 5000$ .

$0.012 (q^{2} - 50 q + 5000)$

Step 2

Use the quadratic formula to find the roots for $q^{2} - 50 q + 5000 = 0$

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Step 2.1

Use the quadratic formula to find the solutions.

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

Step 2.2

Substitute the values $a = 1$ , $b = - 50$ , and $c = 5000$ into the quadratic formula and solve for $q$ .

$\frac{50 \pm \sqrt{{(- 50)}^{2} - 4 \cdot (1 \cdot 5000)}}{2 \cdot 1}$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the numerator.

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Step 2.3.1.1

Raise $- 50$ to the power of $2$ .

$q = \frac{50 \pm \sqrt{2500 - 4 \cdot 1 \cdot 5000}}{2 \cdot 1}$

Step 2.3.1.2

Multiply $- 4 \cdot 1 \cdot 5000$ .

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Step 2.3.1.2.1

Multiply $- 4$ by $1$ .

$q = \frac{50 \pm \sqrt{2500 - 4 \cdot 5000}}{2 \cdot 1}$

Step 2.3.1.2.2

Multiply $- 4$ by $5000$ .

$q = \frac{50 \pm \sqrt{2500 - 20000}}{2 \cdot 1}$

Step 2.3.1.3

Subtract $20000$ from $2500$ .

$q = \frac{50 \pm \sqrt{- 17500}}{2 \cdot 1}$

Step 2.3.1.4

Rewrite $- 17500$ as $- 1 (17500)$ .

$q = \frac{50 \pm \sqrt{- 1 \cdot 17500}}{2 \cdot 1}$

Step 2.3.1.5

Rewrite $\sqrt{- 1 (17500)}$ as $\sqrt{- 1} \cdot \sqrt{17500}$ .

$q = \frac{50 \pm \sqrt{- 1} \cdot \sqrt{17500}}{2 \cdot 1}$

Step 2.3.1.6

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

$q = \frac{50 \pm i \cdot \sqrt{17500}}{2 \cdot 1}$

Step 2.3.1.7

Rewrite $17500$ as $50^{2} \cdot 7$ .

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Step 2.3.1.7.1

Factor $2500$ out of $17500$ .

$q = \frac{50 \pm i \cdot \sqrt{2500 (7)}}{2 \cdot 1}$

Step 2.3.1.7.2

Rewrite $2500$ as $50^{2}$ .

$q = \frac{50 \pm i \cdot \sqrt{50^{2} \cdot 7}}{2 \cdot 1}$

Step 2.3.1.8

Pull terms out from under the radical.

$q = \frac{50 \pm i \cdot (50 \sqrt{7})}{2 \cdot 1}$

Step 2.3.1.9

Move $50$ to the left of $i$ .

$q = \frac{50 \pm 50 i \sqrt{7}}{2 \cdot 1}$

Step 2.3.2

Multiply $2$ by $1$ .

$q = \frac{50 \pm 50 i \sqrt{7}}{2}$

Step 2.3.3

Simplify $\frac{50 \pm 50 i \sqrt{7}}{2}$ .

$q = 25 \pm 25 i \sqrt{7}$

Step 3

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

$0.012 (q - (25 + 25 i \sqrt{7})) (q - (25 - 25 i \sqrt{7}))$