Algebra Examples

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

Step 1

Subtract $61$ from both sides of the equation.

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

Step 2

Subtract $61$ from $39$ .

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $9$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Divide $z$ by $1$ .

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.3.1.2

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

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Step 5.3.3.1.2.1

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

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

Step 5.3.3.1.2.2

Multiply $9$ by $2$ .

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

Step 5.4

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

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

Step 5.5

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

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

Step 5.6

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

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Step 5.6.1

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

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

Step 5.6.2

Simplify the left side.

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Step 5.6.2.1

Cancel the common factor of $9$ .

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Step 5.6.2.1.1

Cancel the common factor.

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

Step 5.6.2.1.2

Divide $z$ by $1$ .

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

Step 5.6.3

Simplify the right side.

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Step 5.6.3.1

Simplify each term.

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Step 5.6.3.1.1

Move the negative in front of the fraction.

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

Step 5.6.3.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.6.3.1.3

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

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Step 5.6.3.1.3.1

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

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

Step 5.6.3.1.3.2

Multiply $9$ by $2$ .

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

Step 5.7

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

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