Finite Math Examples

0.3 x + 0.3 x^{2} + 0.8 x^{3} = 0

$0.3 x + 0.3 x^{2} + 0.8 x^{3} = 0$ ,

$0.5 x + 0.6 x^{2} = 0$ ,

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 1

Factor

$0.1 x$ out of

$0.3 x + 0.3 x^{2} + 0.8 x^{3}$ .

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Step 1.1

Factor

$0.1 x$ out of

$0.3 x$ .

$0.1 x (3) + 0.3 x^{2} + 0.8 x^{3} = 0$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 1.2

Factor

$0.1 x$ out of

$0.3 x^{2}$ .

$0.1 x (3) + 0.1 x (3 x) + 0.8 x^{3} = 0$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 1.3

Factor

$0.1 x$ out of

$0.8 x^{3}$ .

$0.1 x (3) + 0.1 x (3 x) + 0.1 x (8 x^{2}) = 0$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 1.4

Factor

$0.1 x$ out of

$0.1 x (3) + 0.1 x (3 x)$ .

$0.1 x (3 + 3 x) + 0.1 x (8 x^{2}) = 0$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 1.5

Factor

$0.1 x$ out of

$0.1 x (3 + 3 x) + 0.1 x (8 x^{2})$ .

$0.1 x (3 + 3 x + 8 x^{2}) = 0$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

$0.1 x (3 + 3 x + 8 x^{2}) = 0$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 2

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$x = 0$

$3 + 3 x + 8 x^{2} = 0$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 3

Set

$x$ equal to

$0$ .

$x = 0$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4

Set

$3 + 3 x + 8 x^{2}$ equal to

$0$ and solve for

$x$ .

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Step 4.1

Set

$3 + 3 x + 8 x^{2}$ equal to

$0$ .

$3 + 3 x + 8 x^{2} = 0$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2

Solve

$3 + 3 x + 8 x^{2} = 0$ for

$x$ .

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Step 4.2.1

Use the quadratic formula to find the solutions.

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

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.2

Substitute the values

$a = 8$ ,

$b = 3$ , and

$c = 3$ into the quadratic formula and solve for

$x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (8 \cdot 3)}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.3

Simplify.

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Step 4.2.3.1

Simplify the numerator.

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Step 4.2.3.1.1

Raise

$3$ to the power of

$2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 8 \cdot 3}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.3.1.2

Multiply

$- 4 \cdot 8 \cdot 3$ .

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Step 4.2.3.1.2.1

Multiply

$- 4$ by

$8$ .

$x = \frac{- 3 \pm \sqrt{9 - 32 \cdot 3}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.3.1.2.2

Multiply

$- 32$ by

$3$ .

$x = \frac{- 3 \pm \sqrt{9 - 96}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

$x = \frac{- 3 \pm \sqrt{9 - 96}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.3.1.3

Subtract

$96$ from

$9$ .

$x = \frac{- 3 \pm \sqrt{- 87}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.3.1.4

Rewrite

$- 87$ as

$- 1 (87)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 87}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.3.1.5

Rewrite

$\sqrt{- 1 (87)}$ as

$\sqrt{- 1} \cdot \sqrt{87}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{87}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.3.1.6

Rewrite

$\sqrt{- 1}$ as

$i$ .

$x = \frac{- 3 \pm i \sqrt{87}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

$x = \frac{- 3 \pm i \sqrt{87}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.3.2

Multiply

$2$ by

$8$ .

$x = \frac{- 3 \pm i \sqrt{87}}{16}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

$x = \frac{- 3 \pm i \sqrt{87}}{16}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.4

Simplify the expression to solve for the

$+$ portion of the

$\pm$ .

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Step 4.2.4.1

Simplify the numerator.

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Step 4.2.4.1.1

Raise

$3$ to the power of

$2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 8 \cdot 3}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.4.1.2

Multiply

$- 4 \cdot 8 \cdot 3$ .

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Step 4.2.4.1.2.1

Multiply

$- 4$ by

$8$ .

$x = \frac{- 3 \pm \sqrt{9 - 32 \cdot 3}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.4.1.2.2

Multiply

$- 32$ by

$3$ .

$x = \frac{- 3 \pm \sqrt{9 - 96}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

$x = \frac{- 3 \pm \sqrt{9 - 96}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.4.1.3

Subtract

$96$ from

$9$ .

$x = \frac{- 3 \pm \sqrt{- 87}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.4.1.4

Rewrite

$- 87$ as

$- 1 (87)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 87}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.4.1.5

Rewrite

$\sqrt{- 1 (87)}$ as

$\sqrt{- 1} \cdot \sqrt{87}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{87}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.4.1.6

Rewrite

$\sqrt{- 1}$ as

$i$ .

$x = \frac{- 3 \pm i \sqrt{87}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

$x = \frac{- 3 \pm i \sqrt{87}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.4.2

Multiply

$2$ by

$8$ .

$x = \frac{- 3 \pm i \sqrt{87}}{16}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.4.3

Change the

$\pm$ to

$+$ .

$x = \frac{- 3 + i \sqrt{87}}{16}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.4.4

Rewrite

$- 3$ as

$- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + i \sqrt{87}}{16}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.4.5

Factor

$- 1$ out of

$i \sqrt{87}$ .

$x = \frac{- 1 \cdot 3 - (- i \sqrt{87})}{16}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.4.6

Factor

$- 1$ out of

$- 1 (3) - (- i \sqrt{87})$ .

$x = \frac{- 1 (3 - i \sqrt{87})}{16}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.4.7

Move the negative in front of the fraction.

$x = - \frac{3 - i \sqrt{87}}{16}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

$x = - \frac{3 - i \sqrt{87}}{16}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.5

Simplify the expression to solve for the

$-$ portion of the

$\pm$ .

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Step 4.2.5.1

Simplify the numerator.

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Step 4.2.5.1.1

Raise

$3$ to the power of

$2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 8 \cdot 3}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.5.1.2

Multiply

$- 4 \cdot 8 \cdot 3$ .

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Step 4.2.5.1.2.1

Multiply

$- 4$ by

$8$ .

$x = \frac{- 3 \pm \sqrt{9 - 32 \cdot 3}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.5.1.2.2

Multiply

$- 32$ by

$3$ .

$x = \frac{- 3 \pm \sqrt{9 - 96}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

$x = \frac{- 3 \pm \sqrt{9 - 96}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.5.1.3

Subtract

$96$ from

$9$ .

$x = \frac{- 3 \pm \sqrt{- 87}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.5.1.4

Rewrite

$- 87$ as

$- 1 (87)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 87}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.5.1.5

Rewrite

$\sqrt{- 1 (87)}$ as

$\sqrt{- 1} \cdot \sqrt{87}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{87}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.5.1.6

Rewrite

$\sqrt{- 1}$ as

$i$ .

$x = \frac{- 3 \pm i \sqrt{87}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

$x = \frac{- 3 \pm i \sqrt{87}}{2 \cdot 8}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.5.2

Multiply

$2$ by

$8$ .

$x = \frac{- 3 \pm i \sqrt{87}}{16}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.5.3

Change the

$\pm$ to

$-$ .

$x = \frac{- 3 - i \sqrt{87}}{16}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.5.4

Rewrite

$- 3$ as

$- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - i \sqrt{87}}{16}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.5.5

Factor

$- 1$ out of

$- i \sqrt{87}$ .

$x = \frac{- 1 \cdot 3 - (i \sqrt{87})}{16}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.5.6

Factor

$- 1$ out of

$- 1 (3) - (i \sqrt{87})$ .

$x = \frac{- 1 (3 + i \sqrt{87})}{16}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.5.7

Move the negative in front of the fraction.

$x = - \frac{3 + i \sqrt{87}}{16}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

$x = - \frac{3 + i \sqrt{87}}{16}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 4.2.6

The final answer is the combination of both solutions.

$x = - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

$x = - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

$x = - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 5

The final solution is all the values that make

$0.1 x (3 + 3 x + 8 x^{2}) = 0$ true.

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$0.5 x + 0.6 x^{2} = 0$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 6

Factor

$0.1 x$ out of

$0.5 x + 0.6 x^{2}$ .

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Step 6.1

Factor

$0.1 x$ out of

$0.5 x$ .

$0.1 x (5) + 0.6 x^{2} = 0$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 6.2

Factor

$0.1 x$ out of

$0.6 x^{2}$ .

$0.1 x (5) + 0.1 x (6 x) = 0$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 6.3

Factor

$0.1 x$ out of

$0.1 x (5) + 0.1 x (6 x)$ .

$0.1 x (5 + 6 x) = 0$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

$0.1 x (5 + 6 x) = 0$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 7

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$x = 0$

$5 + 6 x = 0$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 8

Set

$x$ equal to

$0$ .

$x = 0$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 9

Set

$5 + 6 x$ equal to

$0$ and solve for

$x$ .

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Step 9.1

Set

$5 + 6 x$ equal to

$0$ .

$5 + 6 x = 0$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 9.2

Solve

$5 + 6 x = 0$ for

$x$ .

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Step 9.2.1

Subtract

$5$ from both sides of the equation.

$6 x = - 5$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 9.2.2

Divide each term in

$6 x = - 5$ by

$6$ and simplify.

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Step 9.2.2.1

Divide each term in

$6 x = - 5$ by

$6$ .

$\frac{6 x}{6} = \frac{- 5}{6}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 9.2.2.2

Simplify the left side.

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Step 9.2.2.2.1

Cancel the common factor of

$6$ .

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Step 9.2.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 5}{6}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 9.2.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{- 5}{6}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

$x = \frac{- 5}{6}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

$x = \frac{- 5}{6}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 9.2.2.3

Simplify the right side.

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Step 9.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{6}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

$x = - \frac{5}{6}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

$x = - \frac{5}{6}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

$x = - \frac{5}{6}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

$x = - \frac{5}{6}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 10

The final solution is all the values that make

$0.1 x (5 + 6 x) = 0$ true.

$x = 0, - \frac{5}{6}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$0.2 x + 0.1 x^{2} + 0.2 x^{3} = 0$

Step 11

Factor

$0.1 x$ out of

$0.2 x + 0.1 x^{2} + 0.2 x^{3}$ .

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Step 11.1

Factor

$0.1 x$ out of

$0.2 x$ .

$0.1 x (2) + 0.1 x^{2} + 0.2 x^{3} = 0$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 11.2

Factor

$0.1 x$ out of

$0.1 x^{2}$ .

$0.1 x (2) + 0.1 x (x) + 0.2 x^{3} = 0$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 11.3

Factor

$0.1 x$ out of

$0.2 x^{3}$ .

$0.1 x (2) + 0.1 x (x) + 0.1 x (2 x^{2}) = 0$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 11.4

Factor

$0.1 x$ out of

$0.1 x (2) + 0.1 x (x)$ .

$0.1 x (2 + x) + 0.1 x (2 x^{2}) = 0$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 11.5

Factor

$0.1 x$ out of

$0.1 x (2 + x) + 0.1 x (2 x^{2})$ .

$0.1 x (2 + x + 2 x^{2}) = 0$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

$0.1 x (2 + x + 2 x^{2}) = 0$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 12

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$x = 0$

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

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 13

Set

$x$ equal to

$0$ .

$x = 0$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14

Set

$2 + x + 2 x^{2}$ equal to

$0$ and solve for

$x$ .

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Step 14.1

Set

$2 + x + 2 x^{2}$ equal to

$0$ .

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

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2

Solve

$2 + x + 2 x^{2} = 0$ for

$x$ .

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Step 14.2.1

Use the quadratic formula to find the solutions.

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

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.2

Substitute the values

$a = 2$ ,

$b = 1$ , and

$c = 2$ into the quadratic formula and solve for

$x$ .

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

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.3

Simplify.

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Step 14.2.3.1

Simplify the numerator.

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Step 14.2.3.1.1

One to any power is one.

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

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.3.1.2

Multiply

$- 4 \cdot 2 \cdot 2$ .

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Step 14.2.3.1.2.1

Multiply

$- 4$ by

$2$ .

$x = \frac{- 1 \pm \sqrt{1 - 8 \cdot 2}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.3.1.2.2

Multiply

$- 8$ by

$2$ .

$x = \frac{- 1 \pm \sqrt{1 - 16}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

$x = \frac{- 1 \pm \sqrt{1 - 16}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.3.1.3

Subtract

$16$ from

$1$ .

$x = \frac{- 1 \pm \sqrt{- 15}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.3.1.4

Rewrite

$- 15$ as

$- 1 (15)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 15}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.3.1.5

Rewrite

$\sqrt{- 1 (15)}$ as

$\sqrt{- 1} \cdot \sqrt{15}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{15}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.3.1.6

Rewrite

$\sqrt{- 1}$ as

$i$ .

$x = \frac{- 1 \pm i \sqrt{15}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

$x = \frac{- 1 \pm i \sqrt{15}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.3.2

Multiply

$2$ by

$2$ .

$x = \frac{- 1 \pm i \sqrt{15}}{4}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

$x = \frac{- 1 \pm i \sqrt{15}}{4}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.4

Simplify the expression to solve for the

$+$ portion of the

$\pm$ .

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Step 14.2.4.1

Simplify the numerator.

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Step 14.2.4.1.1

One to any power is one.

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

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.4.1.2

Multiply

$- 4 \cdot 2 \cdot 2$ .

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Step 14.2.4.1.2.1

Multiply

$- 4$ by

$2$ .

$x = \frac{- 1 \pm \sqrt{1 - 8 \cdot 2}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.4.1.2.2

Multiply

$- 8$ by

$2$ .

$x = \frac{- 1 \pm \sqrt{1 - 16}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

$x = \frac{- 1 \pm \sqrt{1 - 16}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.4.1.3

Subtract

$16$ from

$1$ .

$x = \frac{- 1 \pm \sqrt{- 15}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.4.1.4

Rewrite

$- 15$ as

$- 1 (15)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 15}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.4.1.5

Rewrite

$\sqrt{- 1 (15)}$ as

$\sqrt{- 1} \cdot \sqrt{15}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{15}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.4.1.6

Rewrite

$\sqrt{- 1}$ as

$i$ .

$x = \frac{- 1 \pm i \sqrt{15}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

$x = \frac{- 1 \pm i \sqrt{15}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.4.2

Multiply

$2$ by

$2$ .

$x = \frac{- 1 \pm i \sqrt{15}}{4}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.4.3

Change the

$\pm$ to

$+$ .

$x = \frac{- 1 + i \sqrt{15}}{4}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.4.4

Rewrite

$- 1$ as

$- 1 (1)$ .

$x = \frac{- 1 \cdot 1 + i \sqrt{15}}{4}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.4.5

Factor

$- 1$ out of

$i \sqrt{15}$ .

$x = \frac{- 1 \cdot 1 - (- i \sqrt{15})}{4}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.4.6

Factor

$- 1$ out of

$- 1 (1) - (- i \sqrt{15})$ .

$x = \frac{- 1 (1 - i \sqrt{15})}{4}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.4.7

Move the negative in front of the fraction.

$x = - \frac{1 - i \sqrt{15}}{4}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

$x = - \frac{1 - i \sqrt{15}}{4}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.5

Simplify the expression to solve for the

$-$ portion of the

$\pm$ .

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Step 14.2.5.1

Simplify the numerator.

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Step 14.2.5.1.1

One to any power is one.

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

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.5.1.2

Multiply

$- 4 \cdot 2 \cdot 2$ .

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Step 14.2.5.1.2.1

Multiply

$- 4$ by

$2$ .

$x = \frac{- 1 \pm \sqrt{1 - 8 \cdot 2}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.5.1.2.2

Multiply

$- 8$ by

$2$ .

$x = \frac{- 1 \pm \sqrt{1 - 16}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

$x = \frac{- 1 \pm \sqrt{1 - 16}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.5.1.3

Subtract

$16$ from

$1$ .

$x = \frac{- 1 \pm \sqrt{- 15}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.5.1.4

Rewrite

$- 15$ as

$- 1 (15)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 15}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.5.1.5

Rewrite

$\sqrt{- 1 (15)}$ as

$\sqrt{- 1} \cdot \sqrt{15}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{15}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.5.1.6

Rewrite

$\sqrt{- 1}$ as

$i$ .

$x = \frac{- 1 \pm i \sqrt{15}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

$x = \frac{- 1 \pm i \sqrt{15}}{2 \cdot 2}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.5.2

Multiply

$2$ by

$2$ .

$x = \frac{- 1 \pm i \sqrt{15}}{4}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.5.3

Change the

$\pm$ to

$-$ .

$x = \frac{- 1 - i \sqrt{15}}{4}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.5.4

Rewrite

$- 1$ as

$- 1 (1)$ .

$x = \frac{- 1 \cdot 1 - i \sqrt{15}}{4}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.5.5

Factor

$- 1$ out of

$- i \sqrt{15}$ .

$x = \frac{- 1 \cdot 1 - (i \sqrt{15})}{4}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.5.6

Factor

$- 1$ out of

$- 1 (1) - (i \sqrt{15})$ .

$x = \frac{- 1 (1 + i \sqrt{15})}{4}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.5.7

Move the negative in front of the fraction.

$x = - \frac{1 + i \sqrt{15}}{4}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

$x = - \frac{1 + i \sqrt{15}}{4}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 14.2.6

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

$x = - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

$x = - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 15

The final solution is all the values that make

$0.1 x (2 + x + 2 x^{2}) = 0$ true.

$x = 0, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4}$

$x = 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}$

$x = 0, - \frac{5}{6}$

Step 16

Since the roots of an equation are the points where the solution is

$0$ , set each root as a factor of the equation that equals

$0$ .

$(x - (0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16})) (x - (- 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16})) (x - (0, - \frac{5}{6})) (x - (0, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4})) (x - (- 0, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4})) = 0$

Step 17

Simplify.

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Step 17.1

Subtract

$0$ from

$x$ .

$(x, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}) (x - 0, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}) (x - 0, - \frac{5}{6}) (x - 0, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4}) (x - 0, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4}) = 0$

Step 17.2

Subtract

$0$ from

$x$ .

$(x, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}) (x, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}) (x - 0, - \frac{5}{6}) (x - 0, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4}) (x - 0, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4}) = 0$

Step 17.3

Multiply

$(x, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16}) (x, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16})$ .

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Step 17.3.1

Raise

$(x, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16})$ to the power of

$1$ .

Step 17.3.2

Raise

$(x, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16})$ to the power of

$1$ .

Step 17.3.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(x, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16})}^{1 + 1} (x - 0, - \frac{5}{6}) (x - 0, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4}) (x - 0, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4}) = 0$

Step 17.3.4

Add

$1$ and

$1$ .

${(x, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16})}^{2} (x - 0, - \frac{5}{6}) (x - 0, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4}) (x - 0, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4}) = 0$

Step 17.4

Subtract

$0$ from

$x$ .

${(x, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16})}^{2} (x, - \frac{5}{6}) (x - 0, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4}) (x - 0, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4}) = 0$

Step 17.5

Subtract

$0$ from

$x$ .

${(x, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16})}^{2} (x, - \frac{5}{6}) (x, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4}) (x - 0, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4}) = 0$

Step 17.6

Subtract

$0$ from

$x$ .

${(x, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16})}^{2} (x, - \frac{5}{6}) (x, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4}) (x, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4}) = 0$

Step 17.7

Multiply

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Step 17.7.1

Raise

$(x, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4})$ to the power of

$1$ .

${(x, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16})}^{2} (x, - \frac{5}{6}) ((x, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4}) (x, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4})) = 0$

Step 17.7.2

Raise

$(x, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4})$ to the power of

$1$ .

Step 17.7.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(x, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16})}^{2} (x, - \frac{5}{6}) {(x, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4})}^{1 + 1} = 0$

Step 17.7.4

Add

$1$ and

$1$ .

${(x, - \frac{3 - i \sqrt{87}}{16}, - \frac{3 + i \sqrt{87}}{16})}^{2} (x, - \frac{5}{6}) {(x, - \frac{1 - i \sqrt{15}}{4}, - \frac{1 + i \sqrt{15}}{4})}^{2} = 0$