Finite Math Examples

${(5 k + 1)}^{2} = 27$

Step 1

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

$5 k + 1 = \pm \sqrt{27}$

Step 2

Simplify $\pm \sqrt{27}$ .

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Step 2.1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 2.1.1

Factor $9$ out of $27$ .

$5 k + 1 = \pm \sqrt{9 (3)}$

Step 2.1.2

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

$5 k + 1 = \pm \sqrt{3^{2} \cdot 3}$

Step 2.2

Pull terms out from under the radical.

$5 k + 1 = \pm 3 \sqrt{3}$

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.

$5 k + 1 = 3 \sqrt{3}$

Step 3.2

Subtract $1$ from both sides of the equation.

$5 k = 3 \sqrt{3} - 1$

Step 3.3

Divide each term in $5 k = 3 \sqrt{3} - 1$ by $5$ and simplify.

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Step 3.3.1

Divide each term in $5 k = 3 \sqrt{3} - 1$ by $5$ .

$\frac{5 k}{5} = \frac{3 \sqrt{3}}{5} + \frac{- 1}{5}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $5$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{5 k}{5} = \frac{3 \sqrt{3}}{5} + \frac{- 1}{5}$

Step 3.3.2.1.2

Divide $k$ by $1$ .

$k = \frac{3 \sqrt{3}}{5} + \frac{- 1}{5}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Move the negative in front of the fraction.

$k = \frac{3 \sqrt{3}}{5} - \frac{1}{5}$

Step 3.4

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

$5 k + 1 = - 3 \sqrt{3}$

Step 3.5

Subtract $1$ from both sides of the equation.

$5 k = - 3 \sqrt{3} - 1$

Step 3.6

Divide each term in $5 k = - 3 \sqrt{3} - 1$ by $5$ and simplify.

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Step 3.6.1

Divide each term in $5 k = - 3 \sqrt{3} - 1$ by $5$ .

$\frac{5 k}{5} = \frac{- 3 \sqrt{3}}{5} + \frac{- 1}{5}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $5$ .

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Step 3.6.2.1.1

Cancel the common factor.

$\frac{5 k}{5} = \frac{- 3 \sqrt{3}}{5} + \frac{- 1}{5}$

Step 3.6.2.1.2

Divide $k$ by $1$ .

$k = \frac{- 3 \sqrt{3}}{5} + \frac{- 1}{5}$

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

Move the negative in front of the fraction.

$k = - \frac{3 \sqrt{3}}{5} + \frac{- 1}{5}$

Step 3.6.3.1.2

Move the negative in front of the fraction.

$k = - \frac{3 \sqrt{3}}{5} - \frac{1}{5}$

Step 3.7

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

$k = \frac{3 \sqrt{3}}{5} - \frac{1}{5}, - \frac{3 \sqrt{3}}{5} - \frac{1}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$k = \frac{3 \sqrt{3}}{5} - \frac{1}{5}, - \frac{3 \sqrt{3}}{5} - \frac{1}{5}$

Decimal Form:

$k = 0.83923048 \dots, - 1.23923048 \dots$