Precalculus Examples

${(3 p + 2)}^{2} = 12$

Step 1

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

$3 p + 2 = \pm \sqrt{12}$

Step 2

Simplify $\pm \sqrt{12}$ .

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Step 2.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.1.1

Factor $4$ out of $12$ .

$3 p + 2 = \pm \sqrt{4 (3)}$

Step 2.1.2

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

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

Step 2.2

Pull terms out from under the radical.

$3 p + 2 = \pm 2 \sqrt{3}$

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.

$3 p + 2 = 2 \sqrt{3}$

Step 3.2

Subtract $2$ from both sides of the equation.

$3 p = 2 \sqrt{3} - 2$

Step 3.3

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

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Step 3.3.1

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

$\frac{3 p}{3} = \frac{2 \sqrt{3}}{3} + \frac{- 2}{3}$

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.

$\frac{3 p}{3} = \frac{2 \sqrt{3}}{3} + \frac{- 2}{3}$

Step 3.3.2.1.2

Divide $p$ by $1$ .

$p = \frac{2 \sqrt{3}}{3} + \frac{- 2}{3}$

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.

$p = \frac{2 \sqrt{3}}{3} - \frac{2}{3}$

Step 3.4

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

$3 p + 2 = - 2 \sqrt{3}$

Step 3.5

Subtract $2$ from both sides of the equation.

$3 p = - 2 \sqrt{3} - 2$

Step 3.6

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

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Step 3.6.1

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

$\frac{3 p}{3} = \frac{- 2 \sqrt{3}}{3} + \frac{- 2}{3}$

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.

$\frac{3 p}{3} = \frac{- 2 \sqrt{3}}{3} + \frac{- 2}{3}$

Step 3.6.2.1.2

Divide $p$ by $1$ .

$p = \frac{- 2 \sqrt{3}}{3} + \frac{- 2}{3}$

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

Move the negative in front of the fraction.

$p = - \frac{2 \sqrt{3}}{3} + \frac{- 2}{3}$

Step 3.6.3.1.2

Move the negative in front of the fraction.

$p = - \frac{2 \sqrt{3}}{3} - \frac{2}{3}$

Step 3.7

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

$p = \frac{2 \sqrt{3}}{3} - \frac{2}{3}, - \frac{2 \sqrt{3}}{3} - \frac{2}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$p = \frac{2 \sqrt{3}}{3} - \frac{2}{3}, - \frac{2 \sqrt{3}}{3} - \frac{2}{3}$

Decimal Form:

$p = 0.48803387 \dots, - 1.82136720 \dots$