Finite Math Examples

${(3 x + 1)}^{2} = 18$

Step 1

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

$3 x + 1 = \pm \sqrt{18}$

Step 2

Simplify $\pm \sqrt{18}$ .

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Step 2.1

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 2.1.1

Factor $9$ out of $18$ .

$3 x + 1 = \pm \sqrt{9 (2)}$

Step 2.1.2

Rewrite $9$ as $3^{2}$ .

$3 x + 1 = \pm \sqrt{3^{2} \cdot 2}$

Step 2.2

Pull terms out from under the radical.

$3 x + 1 = \pm 3 \sqrt{2}$

Step 3

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

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Step 3.1

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

$3 x + 1 = 3 \sqrt{2}$

Step 3.2

Subtract $1$ from both sides of the equation.

$3 x = 3 \sqrt{2} - 1$

Step 3.3

Divide each term in $3 x = 3 \sqrt{2} - 1$ by $3$ and simplify.

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Step 3.3.1

Divide each term in $3 x = 3 \sqrt{2} - 1$ by $3$ .

$\frac{3 x}{3} = \frac{3 \sqrt{2}}{3} + \frac{- 1}{3}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $3$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{3 \sqrt{2}}{3} + \frac{- 1}{3}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{3 \sqrt{2}}{3} + \frac{- 1}{3}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Cancel the common factor of $3$ .

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Step 3.3.3.1.1.1

Cancel the common factor.

$x = \frac{3 \sqrt{2}}{3} + \frac{- 1}{3}$

Step 3.3.3.1.1.2

Divide $\sqrt{2}$ by $1$ .

$x = \sqrt{2} + \frac{- 1}{3}$

Step 3.3.3.1.2

Move the negative in front of the fraction.

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

Step 3.4

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

$3 x + 1 = - 3 \sqrt{2}$

Step 3.5

Subtract $1$ from both sides of the equation.

$3 x = - 3 \sqrt{2} - 1$

Step 3.6

Divide each term in $3 x = - 3 \sqrt{2} - 1$ by $3$ and simplify.

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Step 3.6.1

Divide each term in $3 x = - 3 \sqrt{2} - 1$ by $3$ .

$\frac{3 x}{3} = \frac{- 3 \sqrt{2}}{3} + \frac{- 1}{3}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $3$ .

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Step 3.6.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 3 \sqrt{2}}{3} + \frac{- 1}{3}$

Step 3.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3 \sqrt{2}}{3} + \frac{- 1}{3}$

Step 3.6.3

Simplify the right side.

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Step 3.6.3.1

Simplify each term.

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Step 3.6.3.1.1

Cancel the common factor of $- 3$ and $3$ .

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Step 3.6.3.1.1.1

Factor $3$ out of $- 3 \sqrt{2}$ .

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

Step 3.6.3.1.1.2

Cancel the common factors.

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Step 3.6.3.1.1.2.1

Factor $3$ out of $3$ .

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

Step 3.6.3.1.1.2.2

Cancel the common factor.

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

Step 3.6.3.1.1.2.3

Rewrite the expression.

$x = \frac{- \sqrt{2}}{1} + \frac{- 1}{3}$

Step 3.6.3.1.1.2.4

Divide $- \sqrt{2}$ by $1$ .

$x = - \sqrt{2} + \frac{- 1}{3}$

Step 3.6.3.1.2

Move the negative in front of the fraction.

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

Step 3.7

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

$x = \sqrt{2} - \frac{1}{3}, - \sqrt{2} - \frac{1}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{2} - \frac{1}{3}, - \sqrt{2} - \frac{1}{3}$

Decimal Form:

$x = 1.08088022 \dots, - 1.74754689 \dots$