Algebra Examples

\frac{- 12 + \sqrt{- 45}}{24}

$\frac{- 12 + \sqrt{- 45}}{24}$

Step 1

Pull out imaginary unit

$i$ .

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Step 1.1

Rewrite

$- 45$ as

$- 1 (45)$ .

$\frac{- 12 + \sqrt{- 1 (45)}}{24}$

Step 1.2

Rewrite

$\sqrt{- 1 (45)}$ as

$\sqrt{- 1} \cdot \sqrt{45}$ .

$\frac{- 12 + \sqrt{- 1} \cdot \sqrt{45}}{24}$

Step 1.3

Rewrite

$\sqrt{- 1}$ as

$i$ .

$\frac{- 12 + i \cdot \sqrt{45}}{24}$

$\frac{- 12 + i \cdot \sqrt{45}}{24}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite

$45$ as

$3^{2} \cdot 5$ .

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Step 2.1.1

Factor

$9$ out of

$45$ .

$\frac{- 12 + i \cdot \sqrt{9 (5)}}{24}$

Step 2.1.2

Rewrite

$9$ as

$3^{2}$ .

$\frac{- 12 + i \cdot \sqrt{3^{2} \cdot 5}}{24}$

$\frac{- 12 + i \cdot \sqrt{3^{2} \cdot 5}}{24}$

Step 2.2

Pull terms out from under the radical.

$\frac{- 12 + i \cdot (3 \sqrt{5})}{24}$

Step 2.3

Move

$3$ to the left of

$i$ .

$\frac{- 12 + 3 i \sqrt{5}}{24}$

$\frac{- 12 + 3 i \sqrt{5}}{24}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of

$- 12 + 3 i \sqrt{5}$ and

$24$ .

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Step 3.1.1

Factor

$3$ out of

$- 12$ .

$\frac{3 \cdot - 4 + 3 i \sqrt{5}}{24}$

Step 3.1.2

Factor

$3$ out of

$3 i \sqrt{5}$ .

$\frac{3 \cdot - 4 + 3 (i \sqrt{5})}{24}$

Step 3.1.3

Factor

$3$ out of

$3 \cdot - 4 + 3 (i \sqrt{5})$ .

$\frac{3 \cdot (- 4 + i \sqrt{5})}{24}$

Step 3.1.4

Cancel the common factors.

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Step 3.1.4.1

Factor

$3$ out of

$24$ .

$\frac{3 \cdot (- 4 + i \sqrt{5})}{3 (8)}$

Step 3.1.4.2

Cancel the common factor.

$\frac{3 \cdot (- 4 + i \sqrt{5})}{3 \cdot 8}$

Step 3.1.4.3

Rewrite the expression.

$\frac{- 4 + i \sqrt{5}}{8}$

$\frac{- 4 + i \sqrt{5}}{8}$

$\frac{- 4 + i \sqrt{5}}{8}$

Step 3.2

Rewrite

$- 4$ as

$- 1 (4)$ .

$\frac{- 1 (4) + i \sqrt{5}}{8}$

Step 3.3

Factor

$- 1$ out of

$i \sqrt{5}$ .

$\frac{- 1 (4) - (- i \sqrt{5})}{8}$

Step 3.4

Factor

$- 1$ out of

$- 1 (4) - (- i \sqrt{5})$ .

$\frac{- 1 (4 - i \sqrt{5})}{8}$

Step 3.5

Move the negative in front of the fraction.

$- \frac{4 - i \sqrt{5}}{8}$

$- \frac{4 - i \sqrt{5}}{8}$

$\frac{- 12 + \sqrt[]{- 45}}{24}$

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$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$