Algebra Examples

$x = \frac{- 6 + - \sqrt{6^{2} - 4 \cdot 3 \cdot 8}}{2 \cdot 3}$

Step 1

Replace all occurrences of $+$ $-$ with a single $-$ . A plus sign followed by a minus sign has the same mathematical meaning as a single minus sign because $1 \cdot - 1 = - 1$

$x = \frac{- 6 - \sqrt{6^{2} - 4 \cdot 3 \cdot 8}}{2 \cdot 3}$

Step 2

Simplify the numerator.

Tap for more steps...

Step 2.1

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

$x = \frac{- 6 - \sqrt{36 - 4 \cdot 3 \cdot 8}}{2 \cdot 3}$

Step 2.2

Multiply $- 4 \cdot 3 \cdot 8$ .

Tap for more steps...

Step 2.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 6 - \sqrt{36 - 12 \cdot 8}}{2 \cdot 3}$

Step 2.2.2

Multiply $- 12$ by $8$ .

$x = \frac{- 6 - \sqrt{36 - 96}}{2 \cdot 3}$

Step 2.3

Subtract $96$ from $36$ .

$x = \frac{- 6 - \sqrt{- 60}}{2 \cdot 3}$

Step 2.4

Rewrite $- 60$ as $- 1 (60)$ .

$x = \frac{- 6 - \sqrt{- 1 (60)}}{2 \cdot 3}$

Step 2.5

Rewrite $\sqrt{- 1 (60)}$ as $\sqrt{- 1} \cdot \sqrt{60}$ .

$x = \frac{- 6 - (\sqrt{- 1} \cdot \sqrt{60})}{2 \cdot 3}$

Step 2.6

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

$x = \frac{- 6 - (i \cdot \sqrt{60})}{2 \cdot 3}$

Step 2.7

Rewrite $60$ as $2^{2} \cdot 15$ .

Tap for more steps...

Step 2.7.1

Factor $4$ out of $60$ .

$x = \frac{- 6 - (i \cdot \sqrt{4 (15)})}{2 \cdot 3}$

Step 2.7.2

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

$x = \frac{- 6 - (i \cdot \sqrt{2^{2} \cdot 15})}{2 \cdot 3}$

Step 2.8

Pull terms out from under the radical.

$x = \frac{- 6 - (i \cdot (2 \sqrt{15}))}{2 \cdot 3}$

Step 2.9

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

$x = \frac{- 6 - (2 \cdot i \sqrt{15})}{2 \cdot 3}$

Step 2.10

Multiply $2$ by $- 1$ .

$x = \frac{- 6 - 2 i \sqrt{15}}{2 \cdot 3}$

Step 3

Simplify terms.

Tap for more steps...

Step 3.1

Multiply $2$ by $3$ .

$x = \frac{- 6 - 2 i \sqrt{15}}{6}$

Step 3.2

Cancel the common factor of $- 6 - 2 i \sqrt{15}$ and $6$ .

Tap for more steps...

Step 3.2.1

Factor $2$ out of $- 6$ .

$x = \frac{2 (- 3) - 2 i \sqrt{15}}{6}$

Step 3.2.2

Factor $2$ out of $- 2 i \sqrt{15}$ .

$x = \frac{2 (- 3) + 2 (- i \sqrt{15})}{6}$

Step 3.2.3

Factor $2$ out of $2 (- 3) + 2 (- i \sqrt{15})$ .

$x = \frac{2 (- 3 - i \sqrt{15})}{6}$

Step 3.2.4

Cancel the common factors.

Tap for more steps...

Step 3.2.4.1

Factor $2$ out of $6$ .

$x = \frac{2 (- 3 - i \sqrt{15})}{2 \cdot 3}$

Step 3.2.4.2

Cancel the common factor.

$x = \frac{2 (- 3 - i \sqrt{15})}{2 \cdot 3}$

Step 3.2.4.3

Rewrite the expression.

$x = \frac{- 3 - i \sqrt{15}}{3}$

Step 3.3

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 (3) - i \sqrt{15}}{3}$

Step 3.4

Factor $- 1$ out of $- i \sqrt{15}$ .

$x = \frac{- 1 (3) - (i \sqrt{15})}{3}$

Step 3.5

Factor $- 1$ out of $- 1 (3) - (i \sqrt{15})$ .

$x = \frac{- 1 (3 + i \sqrt{15})}{3}$

Step 3.6

Move the negative in front of the fraction.

$x = - \frac{3 + i \sqrt{15}}{3}$