Solve for z (3z-18)^2+60=42

${(3 z - 18)}^{2} + 60 = 42$

Step 1

Subtract $60$ from both sides of the equation.

${(3 z - 18)}^{2} = 42 - 60$

Step 2

Subtract $60$ from $42$ .

${(3 z - 18)}^{2} = - 18$

Step 3

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

$3 z - 18 = \pm \sqrt{- 18}$

Step 4

Simplify $\pm \sqrt{- 18}$ .

Rewrite $- 18$ as $- 1 (18)$ .

$3 z - 18 = \pm \sqrt{- 1 (18)}$

Rewrite $\sqrt{- 1 (18)}$ as $\sqrt{- 1} \cdot \sqrt{18}$ .

$3 z - 18 = \pm \sqrt{- 1} \cdot \sqrt{18}$

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

$3 z - 18 = \pm i \cdot \sqrt{18}$

Rewrite $18$ as $3^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$3 z - 18 = \pm i \cdot \sqrt{9 (2)}$

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

$3 z - 18 = \pm i \cdot \sqrt{3^{2} \cdot 2}$

Pull terms out from under the radical.

$3 z - 18 = \pm i \cdot (3 \sqrt{2})$

Move $3$ to the left of $i$ .

$3 z - 18 = \pm 3 i \sqrt{2}$

Step 5

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.

$3 z - 18 = 3 i \sqrt{2}$

Add $18$ to both sides of the equation.

$3 z = 3 i \sqrt{2} + 18$

Divide each term in $3 z = 3 i \sqrt{2} + 18$ by $3$ and simplify.

Divide each term in $3 z = 3 i \sqrt{2} + 18$ by $3$ .

$\frac{3 z}{3} = \frac{3 i \sqrt{2}}{3} + \frac{18}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 z}{3} = \frac{3 i \sqrt{2}}{3} + \frac{18}{3}$

Divide $z$ by $1$ .

$z = \frac{3 i \sqrt{2}}{3} + \frac{18}{3}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $3$ .

Cancel the common factor.

$z = \frac{3 i \sqrt{2}}{3} + \frac{18}{3}$

Divide $i \sqrt{2}$ by $1$ .

$z = i \sqrt{2} + \frac{18}{3}$

Divide $18$ by $3$ .

$z = i \sqrt{2} + 6$

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

$3 z - 18 = - 3 i \sqrt{2}$

Add $18$ to both sides of the equation.

$3 z = - 3 i \sqrt{2} + 18$

Divide each term in $3 z = - 3 i \sqrt{2} + 18$ by $3$ and simplify.

Divide each term in $3 z = - 3 i \sqrt{2} + 18$ by $3$ .

$\frac{3 z}{3} = \frac{- 3 i \sqrt{2}}{3} + \frac{18}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 z}{3} = \frac{- 3 i \sqrt{2}}{3} + \frac{18}{3}$

Divide $z$ by $1$ .

$z = \frac{- 3 i \sqrt{2}}{3} + \frac{18}{3}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 3$ and $3$ .

Factor $3$ out of $- 3 i \sqrt{2}$ .

$z = \frac{3 (- i \sqrt{2})}{3} + \frac{18}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

$z = \frac{3 (- i \sqrt{2})}{3 (1)} + \frac{18}{3}$

Cancel the common factor.

$z = \frac{3 (- i \sqrt{2})}{3 \cdot 1} + \frac{18}{3}$

Rewrite the expression.

$z = \frac{- i \sqrt{2}}{1} + \frac{18}{3}$

Divide $- i \sqrt{2}$ by $1$ .

$z = - i \sqrt{2} + \frac{18}{3}$

Divide $18$ by $3$ .

$z = - i \sqrt{2} + 6$

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

$z = i \sqrt{2} + 6, - i \sqrt{2} + 6$

Solve for z (3z-18)^2+60=42

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