Finite Math Examples

$s^{2} + 6 s + 36$

Step 1

Use the quadratic formula to find the roots for $s^{2} + 6 s + 36 = 0$

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Step 1.1

Use the quadratic formula to find the solutions.

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

Step 1.2

Substitute the values $a = 1$ , $b = 6$ , and $c = 36$ into the quadratic formula and solve for $s$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot 36)}}{2 \cdot 1}$

Step 1.3

Simplify.

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Step 1.3.1

Simplify the numerator.

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Step 1.3.1.1

Raise $6$ to the power of $2$ .

$s = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 36}}{2 \cdot 1}$

Step 1.3.1.2

Multiply $- 4 \cdot 1 \cdot 36$ .

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Step 1.3.1.2.1

Multiply $- 4$ by $1$ .

$s = \frac{- 6 \pm \sqrt{36 - 4 \cdot 36}}{2 \cdot 1}$

Step 1.3.1.2.2

Multiply $- 4$ by $36$ .

$s = \frac{- 6 \pm \sqrt{36 - 144}}{2 \cdot 1}$

Step 1.3.1.3

Subtract $144$ from $36$ .

$s = \frac{- 6 \pm \sqrt{- 108}}{2 \cdot 1}$

Step 1.3.1.4

Rewrite $- 108$ as $- 1 (108)$ .

$s = \frac{- 6 \pm \sqrt{- 1 \cdot 108}}{2 \cdot 1}$

Step 1.3.1.5

Rewrite $\sqrt{- 1 (108)}$ as $\sqrt{- 1} \cdot \sqrt{108}$ .

$s = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{108}}{2 \cdot 1}$

Step 1.3.1.6

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

$s = \frac{- 6 \pm i \cdot \sqrt{108}}{2 \cdot 1}$

Step 1.3.1.7

Rewrite $108$ as $6^{2} \cdot 3$ .

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Step 1.3.1.7.1

Factor $36$ out of $108$ .

$s = \frac{- 6 \pm i \cdot \sqrt{36 (3)}}{2 \cdot 1}$

Step 1.3.1.7.2

Rewrite $36$ as $6^{2}$ .

$s = \frac{- 6 \pm i \cdot \sqrt{6^{2} \cdot 3}}{2 \cdot 1}$

Step 1.3.1.8

Pull terms out from under the radical.

$s = \frac{- 6 \pm i \cdot (6 \sqrt{3})}{2 \cdot 1}$

Step 1.3.1.9

Move $6$ to the left of $i$ .

$s = \frac{- 6 \pm 6 i \sqrt{3}}{2 \cdot 1}$

Step 1.3.2

Multiply $2$ by $1$ .

$s = \frac{- 6 \pm 6 i \sqrt{3}}{2}$

Step 1.3.3

Simplify $\frac{- 6 \pm 6 i \sqrt{3}}{2}$ .

$s = - 3 \pm 3 i \sqrt{3}$

Step 2

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

$(s - (- 3 + 3 i \sqrt{3})) (s - (- 3 - 3 i \sqrt{3}))$