Algebra Examples

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

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 $1 \cdot - 1 = - 1$

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

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 $\frac{- 8 - \sqrt{8^{2} - 4 \cdot 3 \cdot - 2}}{2 (3)}$ .

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Step 2.1.1.1

Simplify the numerator.

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Step 2.1.1.1.1

Raise $8$ to the power of $2$ .

$x = \frac{- 8 - \sqrt{64 - 4 \cdot 3 \cdot - 2}}{2 (3)}$

Step 2.1.1.1.2

Multiply $- 4 \cdot 3 \cdot - 2$ .

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Step 2.1.1.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 8 - \sqrt{64 - 12 \cdot - 2}}{2 (3)}$

Step 2.1.1.1.2.2

Multiply $- 12$ by $- 2$ .

$x = \frac{- 8 - \sqrt{64 + 24}}{2 (3)}$

Step 2.1.1.1.3

Add $64$ and $24$ .

$x = \frac{- 8 - \sqrt{88}}{2 (3)}$

Step 2.1.1.1.4

Rewrite $88$ as $2^{2} \cdot 22$ .

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Step 2.1.1.1.4.1

Factor $4$ out of $88$ .

$x = \frac{- 8 - \sqrt{4 (22)}}{2 (3)}$

Step 2.1.1.1.4.2

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

$x = \frac{- 8 - \sqrt{2^{2} \cdot 22}}{2 (3)}$

Step 2.1.1.1.5

Pull terms out from under the radical.

$x = \frac{- 8 - (2 \sqrt{22})}{2 (3)}$

Step 2.1.1.1.6

Multiply $2$ by $- 1$ .

$x = \frac{- 8 - 2 \sqrt{22}}{2 (3)}$

Step 2.1.1.2

Simplify terms.

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Step 2.1.1.2.1

Multiply $2$ by $3$ .

$x = \frac{- 8 - 2 \sqrt{22}}{6}$

Step 2.1.1.2.2

Cancel the common factor of $- 8 - 2 \sqrt{22}$ and $6$ .

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Step 2.1.1.2.2.1

Factor $2$ out of $- 8$ .

$x = \frac{2 (- 4) - 2 \sqrt{22}}{6}$

Step 2.1.1.2.2.2

Factor $2$ out of $- 2 \sqrt{22}$ .

$x = \frac{2 (- 4) + 2 (- \sqrt{22})}{6}$

Step 2.1.1.2.2.3

Factor $2$ out of $2 (- 4) + 2 (- \sqrt{22})$ .

$x = \frac{2 (- 4 - \sqrt{22})}{6}$

Step 2.1.1.2.2.4

Cancel the common factors.

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Step 2.1.1.2.2.4.1

Factor $2$ out of $6$ .

$x = \frac{2 (- 4 - \sqrt{22})}{2 \cdot 3}$

Step 2.1.1.2.2.4.2

Cancel the common factor.

$x = \frac{2 (- 4 - \sqrt{22})}{2 \cdot 3}$

Step 2.1.1.2.2.4.3

Rewrite the expression.

$x = \frac{- 4 - \sqrt{22}}{3}$

Step 2.1.1.2.3

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{- 1 (4) - \sqrt{22}}{3}$

Step 2.1.1.2.4

Factor $- 1$ out of $- \sqrt{22}$ .

$x = \frac{- 1 (4) - (\sqrt{22})}{3}$

Step 2.1.1.2.5

Factor $- 1$ out of $- 1 (4) - (\sqrt{22})$ .

$x = \frac{- 1 (4 + \sqrt{22})}{3}$

Step 2.1.1.2.6

Move the negative in front of the fraction.

$x = - \frac{4 + \sqrt{22}}{3}$

Step 2.2

Add $\frac{4 + \sqrt{22}}{3}$ to both sides of the equation.

$x + \frac{4 + \sqrt{22}}{3} = 0$

Step 3

Simplify $x + \frac{4 + \sqrt{22}}{3}$ .

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Step 3.1

To write $x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x \cdot \frac{3}{3} + \frac{4 + \sqrt{22}}{3} = 0$

Step 3.2

Simplify terms.

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Step 3.2.1

Combine $x$ and $\frac{3}{3}$ .

$\frac{x \cdot 3}{3} + \frac{4 + \sqrt{22}}{3} = 0$

Step 3.2.2

Combine the numerators over the common denominator.

$\frac{x \cdot 3 + 4 + \sqrt{22}}{3} = 0$

Step 3.3

Move $3$ to the left of $x$ .

$\frac{3 x + 4 + \sqrt{22}}{3} = 0$

Step 4

Set the numerator equal to zero.

$3 x + 4 + \sqrt{22} = 0$

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 $4$ from both sides of the equation.

$3 x + \sqrt{22} = - 4$

Step 5.1.2

Subtract $\sqrt{22}$ from both sides of the equation.

$3 x = - 4 - \sqrt{22}$

Step 5.2

Divide each term in $3 x = - 4 - \sqrt{22}$ by $3$ and simplify.

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Step 5.2.1

Divide each term in $3 x = - 4 - \sqrt{22}$ by $3$ .

$\frac{3 x}{3} = \frac{- 4}{3} + \frac{- \sqrt{22}}{3}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 4}{3} + \frac{- \sqrt{22}}{3}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{3} + \frac{- \sqrt{22}}{3}$

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.

$x = - \frac{4}{3} + \frac{- \sqrt{22}}{3}$

Step 5.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{4}{3} - \frac{\sqrt{22}}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{4}{3} - \frac{\sqrt{22}}{3}$

Decimal Form:

$x = - 2.89680525 \dots$