Solve Using the Square Root Property (4y+22)^2=27

${(4 y + 22)}^{2} = 27$

Step 1

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

$4 y + 22 = \pm \sqrt{27}$

Step 2

Simplify $\pm \sqrt{27}$ .

Rewrite $27$ as $3^{2} \cdot 3$ .

Factor $9$ out of $27$ .

$4 y + 22 = \pm \sqrt{9 (3)}$

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

$4 y + 22 = \pm \sqrt{3^{2} \cdot 3}$

Pull terms out from under the radical.

$4 y + 22 = \pm 3 \sqrt{3}$

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.

$4 y + 22 = 3 \sqrt{3}$

Subtract $22$ from both sides of the equation.

$4 y = 3 \sqrt{3} - 22$

Divide each term in $4 y = 3 \sqrt{3} - 22$ by $4$ and simplify.

Divide each term in $4 y = 3 \sqrt{3} - 22$ by $4$ .

$\frac{4 y}{4} = \frac{3 \sqrt{3}}{4} + \frac{- 22}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y}{4} = \frac{3 \sqrt{3}}{4} + \frac{- 22}{4}$

Divide $y$ by $1$ .

$y = \frac{3 \sqrt{3}}{4} + \frac{- 22}{4}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 22$ and $4$ .

Factor $2$ out of $- 22$ .

$y = \frac{3 \sqrt{3}}{4} + \frac{2 (- 11)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$y = \frac{3 \sqrt{3}}{4} + \frac{2 \cdot - 11}{2 \cdot 2}$

Cancel the common factor.

$y = \frac{3 \sqrt{3}}{4} + \frac{2 \cdot - 11}{2 \cdot 2}$

Rewrite the expression.

$y = \frac{3 \sqrt{3}}{4} + \frac{- 11}{2}$

Move the negative in front of the fraction.

$y = \frac{3 \sqrt{3}}{4} - \frac{11}{2}$

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

$4 y + 22 = - 3 \sqrt{3}$

Subtract $22$ from both sides of the equation.

$4 y = - 3 \sqrt{3} - 22$

Divide each term in $4 y = - 3 \sqrt{3} - 22$ by $4$ and simplify.

Divide each term in $4 y = - 3 \sqrt{3} - 22$ by $4$ .

$\frac{4 y}{4} = \frac{- 3 \sqrt{3}}{4} + \frac{- 22}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y}{4} = \frac{- 3 \sqrt{3}}{4} + \frac{- 22}{4}$

Divide $y$ by $1$ .

$y = \frac{- 3 \sqrt{3}}{4} + \frac{- 22}{4}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$y = - \frac{3 \sqrt{3}}{4} + \frac{- 22}{4}$

Cancel the common factor of $- 22$ and $4$ .

Factor $2$ out of $- 22$ .

$y = - \frac{3 \sqrt{3}}{4} + \frac{2 (- 11)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$y = - \frac{3 \sqrt{3}}{4} + \frac{2 \cdot - 11}{2 \cdot 2}$

Cancel the common factor.

$y = - \frac{3 \sqrt{3}}{4} + \frac{2 \cdot - 11}{2 \cdot 2}$

Rewrite the expression.

$y = - \frac{3 \sqrt{3}}{4} + \frac{- 11}{2}$

Move the negative in front of the fraction.

$y = - \frac{3 \sqrt{3}}{4} - \frac{11}{2}$

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

$y = \frac{3 \sqrt{3}}{4} - \frac{11}{2}, - \frac{3 \sqrt{3}}{4} - \frac{11}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$y = \frac{3 \sqrt{3}}{4} - \frac{11}{2}, - \frac{3 \sqrt{3}}{4} - \frac{11}{2}$

Decimal Form:

$y = - 4.20096189 \dots, - 6.79903810 \dots$

Solve Using the Square Root Property (4y+22)^2=27

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