Algebra Examples

$x = \frac{- 3 + - \sqrt{3^{2} - 4 \cdot 1 \cdot 24}}{2 (1)}$

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 $1 \cdot - 1 = - 1$

$x = \frac{- 3 - \sqrt{3^{2} - 4 \cdot 1 \cdot 24}}{2 (1)}$

Step 2

Move all terms to the left side of the equation and simplify.

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Step 2.1

Simplify the right side.

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Step 2.1.1

Simplify $\frac{- 3 - \sqrt{3^{2} - 4 \cdot 1 \cdot 24}}{2 (1)}$ .

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Step 2.1.1.1

Simplify the numerator.

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Step 2.1.1.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 - \sqrt{9 - 4 \cdot 1 \cdot 24}}{2 (1)}$

Step 2.1.1.1.2

Multiply $- 4 \cdot 1 \cdot 24$ .

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Step 2.1.1.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 - \sqrt{9 - 4 \cdot 24}}{2 (1)}$

Step 2.1.1.1.2.2

Multiply $- 4$ by $24$ .

$x = \frac{- 3 - \sqrt{9 - 96}}{2 (1)}$

Step 2.1.1.1.3

Subtract $96$ from $9$ .

$x = \frac{- 3 - \sqrt{- 87}}{2 (1)}$

Step 2.1.1.1.4

Rewrite $- 87$ as $- 1 (87)$ .

$x = \frac{- 3 - \sqrt{- 1 (87)}}{2 (1)}$

Step 2.1.1.1.5

Rewrite $\sqrt{- 1 (87)}$ as $\sqrt{- 1} \cdot \sqrt{87}$ .

$x = \frac{- 3 - (\sqrt{- 1} \cdot \sqrt{87})}{2 (1)}$

Step 2.1.1.1.6

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

$x = \frac{- 3 - i \sqrt{87}}{2 (1)}$

Step 2.1.1.2

Simplify with factoring out.

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Step 2.1.1.2.1

Multiply $2$ by $1$ .

$x = \frac{- 3 - i \sqrt{87}}{2}$

Step 2.1.1.2.2

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

$x = \frac{- 1 (3) - i \sqrt{87}}{2}$

Step 2.1.1.2.3

Factor $- 1$ out of $- i \sqrt{87}$ .

$x = \frac{- 1 (3) - (i \sqrt{87})}{2}$

Step 2.1.1.2.4

Factor $- 1$ out of $- 1 (3) - (i \sqrt{87})$ .

$x = \frac{- 1 (3 + i \sqrt{87})}{2}$

Step 2.1.1.2.5

Move the negative in front of the fraction.

$x = - \frac{3 + i \sqrt{87}}{2}$

Step 2.2

Add $\frac{3 + i \sqrt{87}}{2}$ to both sides of the equation.

$x + \frac{3 + i \sqrt{87}}{2} = 0$

Step 3

Simplify $x + \frac{3 + i \sqrt{87}}{2}$ .

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Step 3.1

To write $x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x \cdot \frac{2}{2} + \frac{3 + i \sqrt{87}}{2} = 0$

Step 3.2

Simplify terms.

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Step 3.2.1

Combine $x$ and $\frac{2}{2}$ .

$\frac{x \cdot 2}{2} + \frac{3 + i \sqrt{87}}{2} = 0$

Step 3.2.2

Combine the numerators over the common denominator.

$\frac{x \cdot 2 + 3 + i \sqrt{87}}{2} = 0$

Step 3.3

Simplify the numerator.

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Step 3.3.1

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

$\frac{2 \cdot x + 3 + i \sqrt{87}}{2} = 0$

Step 3.3.2

Reorder terms.

$\frac{i \sqrt{87} + 2 x + 3}{2} = 0$

Step 4

Set the numerator equal to zero.

$i \sqrt{87} + 2 x + 3 = 0$

Step 5

Solve the equation for $x$ .

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Step 5.1

Move all terms not containing $x$ to the right side of the equation.

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Step 5.1.1

Subtract $i \sqrt{87}$ from both sides of the equation.

$2 x + 3 = - i \sqrt{87}$

Step 5.1.2

Subtract $3$ from both sides of the equation.

$2 x = - i \sqrt{87} - 3$

Step 5.1.3

Reorder $- i \sqrt{87}$ and $- 3$ .

$2 x = - 3 - i \sqrt{87}$

Step 5.2

Divide each term in $2 x = - 3 - i \sqrt{87}$ by $2$ and simplify.

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Step 5.2.1

Divide each term in $2 x = - 3 - i \sqrt{87}$ by $2$ .

$\frac{2 x}{2} = \frac{- 3}{2} + \frac{- i \sqrt{87}}{2}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 3}{2} + \frac{- i \sqrt{87}}{2}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{2} + \frac{- i \sqrt{87}}{2}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Move the negative in front of the fraction.

$x = - \frac{3}{2} + \frac{- i \sqrt{87}}{2}$

Step 5.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{3}{2} - \frac{i \sqrt{87}}{2}$