Trigonometry Examples

${(6 x + 4)}^{2} = 77$

Step 1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$6 x + 4 = \pm \sqrt{77}$

Step 2

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.1

First, use the positive value of the $\pm$ to find the first solution.

$6 x + 4 = \sqrt{77}$

Step 2.2

Subtract $4$ from both sides of the equation.

$6 x = \sqrt{77} - 4$

Step 2.3

Divide each term in $6 x = \sqrt{77} - 4$ by $6$ and simplify.

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Step 2.3.1

Divide each term in $6 x = \sqrt{77} - 4$ by $6$ .

$\frac{6 x}{6} = \frac{\sqrt{77}}{6} + \frac{- 4}{6}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $6$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{\sqrt{77}}{6} + \frac{- 4}{6}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\sqrt{77}}{6} + \frac{- 4}{6}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

Cancel the common factor of $- 4$ and $6$ .

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Step 2.3.3.1.1.1

Factor $2$ out of $- 4$ .

$x = \frac{\sqrt{77}}{6} + \frac{2 (- 2)}{6}$

Step 2.3.3.1.1.2

Cancel the common factors.

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Step 2.3.3.1.1.2.1

Factor $2$ out of $6$ .

$x = \frac{\sqrt{77}}{6} + \frac{2 \cdot - 2}{2 \cdot 3}$

Step 2.3.3.1.1.2.2

Cancel the common factor.

$x = \frac{\sqrt{77}}{6} + \frac{2 \cdot - 2}{2 \cdot 3}$

Step 2.3.3.1.1.2.3

Rewrite the expression.

$x = \frac{\sqrt{77}}{6} + \frac{- 2}{3}$

Step 2.3.3.1.2

Move the negative in front of the fraction.

$x = \frac{\sqrt{77}}{6} - \frac{2}{3}$

Step 2.4

Next, use the negative value of the $\pm$ to find the second solution.

$6 x + 4 = - \sqrt{77}$

Step 2.5

Subtract $4$ from both sides of the equation.

$6 x = - \sqrt{77} - 4$

Step 2.6

Divide each term in $6 x = - \sqrt{77} - 4$ by $6$ and simplify.

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Step 2.6.1

Divide each term in $6 x = - \sqrt{77} - 4$ by $6$ .

$\frac{6 x}{6} = \frac{- \sqrt{77}}{6} + \frac{- 4}{6}$

Step 2.6.2

Simplify the left side.

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Step 2.6.2.1

Cancel the common factor of $6$ .

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Step 2.6.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{- \sqrt{77}}{6} + \frac{- 4}{6}$

Step 2.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \sqrt{77}}{6} + \frac{- 4}{6}$

Step 2.6.3

Simplify the right side.

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Step 2.6.3.1

Simplify each term.

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Step 2.6.3.1.1

Move the negative in front of the fraction.

$x = - \frac{\sqrt{77}}{6} + \frac{- 4}{6}$

Step 2.6.3.1.2

Cancel the common factor of $- 4$ and $6$ .

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Step 2.6.3.1.2.1

Factor $2$ out of $- 4$ .

$x = - \frac{\sqrt{77}}{6} + \frac{2 (- 2)}{6}$

Step 2.6.3.1.2.2

Cancel the common factors.

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Step 2.6.3.1.2.2.1

Factor $2$ out of $6$ .

$x = - \frac{\sqrt{77}}{6} + \frac{2 \cdot - 2}{2 \cdot 3}$

Step 2.6.3.1.2.2.2

Cancel the common factor.

$x = - \frac{\sqrt{77}}{6} + \frac{2 \cdot - 2}{2 \cdot 3}$

Step 2.6.3.1.2.2.3

Rewrite the expression.

$x = - \frac{\sqrt{77}}{6} + \frac{- 2}{3}$

Step 2.6.3.1.3

Move the negative in front of the fraction.

$x = - \frac{\sqrt{77}}{6} - \frac{2}{3}$

Step 2.7

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{77}}{6} - \frac{2}{3}, - \frac{\sqrt{77}}{6} - \frac{2}{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{77}}{6} - \frac{2}{3}, - \frac{\sqrt{77}}{6} - \frac{2}{3}$

Decimal Form:

$x = 0.79582739 \dots, - 2.12916073 \dots$