Finite Math Examples

$z^{2} - 2 i z + 1 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 1$ , $b = - 2 i$ , and $c = 1$ into the quadratic formula and solve for $z$ .

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

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Apply the product rule to $- 2 i$ .

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

Step 3.1.2

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

$z = \frac{2 i \pm \sqrt{4 i^{2} - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 3.1.3

Rewrite $i^{2}$ as $- 1$ .

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

Step 3.1.4

Multiply $4$ by $- 1$ .

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

Step 3.1.5

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

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Step 3.1.5.1

Multiply $- 4$ by $1$ .

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

Step 3.1.5.2

Multiply $- 4$ by $1$ .

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

Step 3.1.6

Subtract $4$ from $- 4$ .

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

Step 3.1.7

Rewrite $- 8$ as $- 1 (8)$ .

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

Step 3.1.8

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

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

Step 3.1.9

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

$z = \frac{2 i \pm i \cdot \sqrt{8}}{2 \cdot 1}$

Step 3.1.10

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.1.10.1

Factor $4$ out of $8$ .

$z = \frac{2 i \pm i \cdot \sqrt{4 (2)}}{2 \cdot 1}$

Step 3.1.10.2

Rewrite $4$ as $2^{2}$ .

$z = \frac{2 i \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 3.1.11

Pull terms out from under the radical.

$z = \frac{2 i \pm i \cdot (2 \sqrt{2})}{2 \cdot 1}$

Step 3.1.12

Move $2$ to the left of $i$ .

$z = \frac{2 i \pm 2 i \sqrt{2}}{2 \cdot 1}$

Step 3.2

Multiply $2$ by $1$ .

$z = \frac{2 i \pm 2 i \sqrt{2}}{2}$

Step 3.3

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

$z = i \pm i \sqrt{2}$

Step 4

The final answer is the combination of both solutions.

$z = i + i \sqrt{2}, i - i \sqrt{2}$