Solve Using the Square Root Property (9z+1/2)^2+61=39

${(9 z + \frac{1}{2})}^{2} + 61 = 39$

Subtract $61$ from both sides of the equation.

${(9 z + \frac{1}{2})}^{2} = 39 - 61$

Subtract $61$ from $39$ .

${(9 z + \frac{1}{2})}^{2} = - 22$

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

$9 z + \frac{1}{2} = \pm \sqrt{- 22}$

Simplify $\pm \sqrt{- 22}$ .

Rewrite $- 22$ as $- 1 (22)$ .

$9 z + \frac{1}{2} = \pm \sqrt{- 1 (22)}$

Rewrite $\sqrt{- 1 (22)}$ as $\sqrt{- 1} \cdot \sqrt{22}$ .

$9 z + \frac{1}{2} = \pm \sqrt{- 1} \cdot \sqrt{22}$

Rewrite $\sqrt{- 1}$ as $i$ .

$9 z + \frac{1}{2} = \pm i \sqrt{22}$

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.

$9 z + \frac{1}{2} = i \sqrt{22}$

Subtract $\frac{1}{2}$ from both sides of the equation.

$9 z = i \sqrt{22} - \frac{1}{2}$

Divide each term in $9 z = i \sqrt{22} - \frac{1}{2}$ by $9$ and simplify.

Divide each term in $9 z = i \sqrt{22} - \frac{1}{2}$ by $9$ .

$\frac{9 z}{9} = \frac{i \sqrt{22}}{9} + \frac{- \frac{1}{2}}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 z}{9} = \frac{i \sqrt{22}}{9} + \frac{- \frac{1}{2}}{9}$

Divide $z$ by $1$ .

$z = \frac{i \sqrt{22}}{9} + \frac{- \frac{1}{2}}{9}$

Simplify the right side.

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

$z = \frac{i \sqrt{22}}{9} - \frac{1}{2} \cdot \frac{1}{9}$

Multiply $- \frac{1}{2} \cdot \frac{1}{9}$ .

Multiply $\frac{1}{9}$ by $\frac{1}{2}$ .

$z = \frac{i \sqrt{22}}{9} - \frac{1}{9 \cdot 2}$

Multiply $9$ by $2$ .

$z = \frac{i \sqrt{22}}{9} - \frac{1}{18}$

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

$9 z + \frac{1}{2} = - i \sqrt{22}$

Subtract $\frac{1}{2}$ from both sides of the equation.

$9 z = - i \sqrt{22} - \frac{1}{2}$

Divide each term in $9 z = - i \sqrt{22} - \frac{1}{2}$ by $9$ and simplify.

Divide each term in $9 z = - i \sqrt{22} - \frac{1}{2}$ by $9$ .

$\frac{9 z}{9} = \frac{- i \sqrt{22}}{9} + \frac{- \frac{1}{2}}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 z}{9} = \frac{- i \sqrt{22}}{9} + \frac{- \frac{1}{2}}{9}$

Divide $z$ by $1$ .

$z = \frac{- i \sqrt{22}}{9} + \frac{- \frac{1}{2}}{9}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$z = - \frac{i \sqrt{22}}{9} + \frac{- \frac{1}{2}}{9}$

Multiply the numerator by the reciprocal of the denominator.

$z = - \frac{i \sqrt{22}}{9} - \frac{1}{2} \cdot \frac{1}{9}$

Multiply $- \frac{1}{2} \cdot \frac{1}{9}$ .

Multiply $\frac{1}{9}$ by $\frac{1}{2}$ .

$z = - \frac{i \sqrt{22}}{9} - \frac{1}{9 \cdot 2}$

Multiply $9$ by $2$ .

$z = - \frac{i \sqrt{22}}{9} - \frac{1}{18}$

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

$z = \frac{i \sqrt{22}}{9} - \frac{1}{18}, - \frac{i \sqrt{22}}{9} - \frac{1}{18}$

Solve Using the Square Root Property (9z+1/2)^2+61=39

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