Precalculus Examples

$x^{2} = - i$

Step 1

Add $i$ to both sides of the equation.

$x^{2} + i = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 1$ , $b = 0$ , and $c = i$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (1 \cdot i)}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raising $0$ to any positive power yields $0$ .

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 1 \cdot i}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4$ by $1$ .

$x = \frac{0 \pm \sqrt{0 - 4 \cdot i}}{2 \cdot 1}$

Step 4.1.3

Subtract $4 i$ from $0$ .

$x = \frac{0 \pm \sqrt{- 4 i}}{2 \cdot 1}$

Step 4.1.4

Rewrite $- 4$ as ${(2 i)}^{2}$ .

$x = \frac{0 \pm \sqrt{{(2 i)}^{2} i}}{2 \cdot 1}$

Step 4.1.5

Pull terms out from under the radical.

$x = \frac{0 \pm 2 i \sqrt{i}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{0 \pm 2 i \sqrt{i}}{2}$

Step 4.3

Simplify $\frac{0 \pm 2 i \sqrt{i}}{2}$ .

$x = \pm i \sqrt{i}$