Algebra Examples

$x = \frac{- (8) + - \sqrt{{(8)}^{2} - 4 \cdot - 3 \cdot 7}}{2 (- 3)}$

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 $1 \cdot - 1 = - 1$

$x = \frac{- (8) - \sqrt{{(8)}^{2} - 4 \cdot - 3 \cdot 7}}{2 (- 3)}$

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 $\frac{- (8) - \sqrt{{(8)}^{2} - 4 \cdot - 3 \cdot 7}}{2 (- 3)}$ .

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Step 2.1.1.1

Simplify the numerator.

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Step 2.1.1.1.1

Multiply $- 1$ by $8$ .

$x = \frac{- 8 - \sqrt{8^{2} - 4 \cdot - 3 \cdot 7}}{2 (- 3)}$

Step 2.1.1.1.2

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

$x = \frac{- 8 - \sqrt{64 - 4 \cdot - 3 \cdot 7}}{2 (- 3)}$

Step 2.1.1.1.3

Multiply $- 4 \cdot - 3 \cdot 7$ .

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Step 2.1.1.1.3.1

Multiply $- 4$ by $- 3$ .

$x = \frac{- 8 - \sqrt{64 + 12 \cdot 7}}{2 (- 3)}$

Step 2.1.1.1.3.2

Multiply $12$ by $7$ .

$x = \frac{- 8 - \sqrt{64 + 84}}{2 (- 3)}$

Step 2.1.1.1.4

Add $64$ and $84$ .

$x = \frac{- 8 - \sqrt{148}}{2 (- 3)}$

Step 2.1.1.1.5

Rewrite $148$ as $2^{2} \cdot 37$ .

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Step 2.1.1.1.5.1

Factor $4$ out of $148$ .

$x = \frac{- 8 - \sqrt{4 (37)}}{2 (- 3)}$

Step 2.1.1.1.5.2

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

$x = \frac{- 8 - \sqrt{2^{2} \cdot 37}}{2 (- 3)}$

Step 2.1.1.1.6

Pull terms out from under the radical.

$x = \frac{- 8 - (2 \sqrt{37})}{2 (- 3)}$

Step 2.1.1.1.7

Multiply $2$ by $- 1$ .

$x = \frac{- 8 - 2 \sqrt{37}}{2 (- 3)}$

Step 2.1.1.2

Simplify terms.

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Step 2.1.1.2.1

Multiply $2$ by $- 3$ .

$x = \frac{- 8 - 2 \sqrt{37}}{- 6}$

Step 2.1.1.2.2

Cancel the common factor of $- 8 - 2 \sqrt{37}$ and $- 6$ .

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Step 2.1.1.2.2.1

Factor $2$ out of $- 8$ .

$x = \frac{2 (- 4) - 2 \sqrt{37}}{- 6}$

Step 2.1.1.2.2.2

Factor $2$ out of $- 2 \sqrt{37}$ .

$x = \frac{2 (- 4) + 2 (- \sqrt{37})}{- 6}$

Step 2.1.1.2.2.3

Factor $2$ out of $2 (- 4) + 2 (- \sqrt{37})$ .

$x = \frac{2 (- 4 - \sqrt{37})}{- 6}$

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 $- 6$ .

$x = \frac{2 (- 4 - \sqrt{37})}{2 \cdot - 3}$

Step 2.1.1.2.2.4.2

Cancel the common factor.

$x = \frac{2 (- 4 - \sqrt{37})}{2 \cdot - 3}$

Step 2.1.1.2.2.4.3

Rewrite the expression.

$x = \frac{- 4 - \sqrt{37}}{- 3}$

Step 2.1.1.2.3

Move the negative in front of the fraction.

$x = - \frac{- 4 - \sqrt{37}}{3}$

Step 2.1.1.2.4

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

$x = - \frac{- 1 (4) - \sqrt{37}}{3}$

Step 2.1.1.2.5

Factor $- 1$ out of $- \sqrt{37}$ .

$x = - \frac{- 1 (4) - (\sqrt{37})}{3}$

Step 2.1.1.2.6

Factor $- 1$ out of $- 1 (4) - (\sqrt{37})$ .

$x = - \frac{- 1 (4 + \sqrt{37})}{3}$

Step 2.1.1.2.7

Simplify the expression.

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Step 2.1.1.2.7.1

Move the negative in front of the fraction.

$x = - - \frac{4 + \sqrt{37}}{3}$

Step 2.1.1.2.7.2

Multiply $- 1$ by $- 1$ .

$x = 1 \frac{4 + \sqrt{37}}{3}$

Step 2.1.1.2.7.3

Multiply $\frac{4 + \sqrt{37}}{3}$ by $1$ .

$x = \frac{4 + \sqrt{37}}{3}$

Step 2.2

Subtract $\frac{4 + \sqrt{37}}{3}$ from both sides of the equation.

$x - \frac{4 + \sqrt{37}}{3} = 0$

Step 3

Simplify $x - \frac{4 + \sqrt{37}}{3}$ .

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Step 3.1

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

$x \cdot \frac{3}{3} - \frac{4 + \sqrt{37}}{3} = 0$

Step 3.2

Simplify terms.

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Step 3.2.1

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

$\frac{x \cdot 3}{3} - \frac{4 + \sqrt{37}}{3} = 0$

Step 3.2.2

Combine the numerators over the common denominator.

$\frac{x \cdot 3 - (4 + \sqrt{37})}{3} = 0$

Step 3.3

Simplify the numerator.

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Step 3.3.1

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

$\frac{3 \cdot x - (4 + \sqrt{37})}{3} = 0$

Step 3.3.2

Apply the distributive property.

$\frac{3 x - 1 \cdot 4 - \sqrt{37}}{3} = 0$

Step 3.3.3

Multiply $- 1$ by $4$ .

$\frac{3 x - 4 - \sqrt{37}}{3} = 0$

Step 4

Set the numerator equal to zero.

$3 x - 4 - \sqrt{37} = 0$

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

Add $4$ to both sides of the equation.

$3 x - \sqrt{37} = 4$

Step 5.1.2

Add $\sqrt{37}$ to both sides of the equation.

$3 x = 4 + \sqrt{37}$

Step 5.2

Divide each term in $3 x = 4 + \sqrt{37}$ by $3$ and simplify.

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Step 5.2.1

Divide each term in $3 x = 4 + \sqrt{37}$ by $3$ .

$\frac{3 x}{3} = \frac{4}{3} + \frac{\sqrt{37}}{3}$

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.

$\frac{3 x}{3} = \frac{4}{3} + \frac{\sqrt{37}}{3}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{3} + \frac{\sqrt{37}}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{3} + \frac{\sqrt{37}}{3}$

Decimal Form:

$x = 3.36092084 \dots$