Solve Using the Square Root Property (3p+1)^2=8

${(3 p + 1)}^{2} = 8$

Step 1

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

$3 p + 1 = \pm \sqrt{8}$

Step 2

Simplify $\pm \sqrt{8}$ .

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$3 p + 1 = \pm \sqrt{4 (2)}$

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

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

Pull terms out from under the radical.

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

Step 3

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

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

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

Subtract $1$ from both sides of the equation.

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

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $p$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

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

Subtract $1$ from both sides of the equation.

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

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $p$ by $1$ .

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

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

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

Move the negative in front of the fraction.

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

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

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$p = 0.60947570 \dots, - 1.27614237 \dots$

Solve Using the Square Root Property (3p+1)^2=8

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