Precalculus Examples

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

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 $a = i$ , $b = - 2$ , and $c = i$ into the quadratic formula and solve for $x$ .

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Multiply the numerator and denominator of $\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (i \cdot i)}}{2 i}$ by the conjugate of $2 i$ to make the denominator real.

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

Step 3.2

Multiply.

Tap for more steps...

Step 3.2.1

Combine.

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

Step 3.2.2

Simplify the numerator.

Tap for more steps...

Step 3.2.2.1

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

$x = \frac{(2 \pm \sqrt{4 - 4 \cdot (i i)}) i}{2 i i}$

Step 3.2.2.2

Multiply $i i$ .

Tap for more steps...

Step 3.2.2.2.1

Raise $i$ to the power of $1$ .

$x = \frac{(2 \pm \sqrt{4 - 4 \cdot (i i)}) i}{2 i i}$

Step 3.2.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.2.2.2.3

Add $1$ and $1$ .

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

Step 3.2.2.3

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

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

Step 3.2.2.4

Multiply $- 4$ by $- 1$ .

$x = \frac{(2 \pm \sqrt{4 + 4}) i}{2 i i}$

Step 3.2.2.5

Add $4$ and $4$ .

$x = \frac{(2 \pm \sqrt{8}) i}{2 i i}$

Step 3.2.2.6

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

Tap for more steps...

Step 3.2.2.6.1

Factor $4$ out of $8$ .

$x = \frac{(2 \pm \sqrt{4 (2)}) i}{2 i i}$

Step 3.2.2.6.2

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

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

Step 3.2.2.7

Pull terms out from under the radical.

$x = \frac{(2 \pm 2 \sqrt{2}) i}{2 i i}$

Step 3.2.3

Simplify the denominator.

Tap for more steps...

Step 3.2.3.1

Add parentheses.

$x = \frac{(2 \pm 2 \sqrt{2}) i}{2 (i i)}$

Step 3.2.3.2

Raise $i$ to the power of $1$ .

$x = \frac{(2 \pm 2 \sqrt{2}) i}{2 (i i)}$

Step 3.2.3.3

Raise $i$ to the power of $1$ .

$x = \frac{(2 \pm 2 \sqrt{2}) i}{2 (i i)}$

Step 3.2.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.2.3.5

Add $1$ and $1$ .

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

Step 3.2.3.6

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

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

Step 3.3

Multiply $2$ by $- 1$ .

$x = \frac{(2 \pm 2 \sqrt{2}) i}{- 2}$

Step 3.4

Move the negative in front of the fraction.

$x = - \frac{(2 \pm 2 \sqrt{2}) i}{2}$

Step 4

The final answer is the combination of both solutions.

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