Solve Using the Square Root Property (4y-3)^2=72

${(4 y - 3)}^{2} = 72$

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

$4 y - 3 = \pm \sqrt{72}$

Simplify $\pm \sqrt{72}$ .

Rewrite $72$ as $6^{2} \cdot 2$ .

Factor $36$ out of $72$ .

$4 y - 3 = \pm \sqrt{36 (2)}$

Rewrite $36$ as $6^{2}$ .

$4 y - 3 = \pm \sqrt{6^{2} \cdot 2}$

Pull terms out from under the radical.

$4 y - 3 = \pm 6 \sqrt{2}$

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.

$4 y - 3 = 6 \sqrt{2}$

Add $3$ to both sides of the equation.

$4 y = 6 \sqrt{2} + 3$

Divide each term in $4 y = 6 \sqrt{2} + 3$ by $4$ and simplify.

Divide each term in $4 y = 6 \sqrt{2} + 3$ by $4$ .

$\frac{4 y}{4} = \frac{6 \sqrt{2}}{4} + \frac{3}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y}{4} = \frac{6 \sqrt{2}}{4} + \frac{3}{4}$

Divide $y$ by $1$ .

$y = \frac{6 \sqrt{2}}{4} + \frac{3}{4}$

Simplify the right side.

Cancel the common factor of $6$ and $4$ .

Factor $2$ out of $6 \sqrt{2}$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

$4 y - 3 = - 6 \sqrt{2}$

Add $3$ to both sides of the equation.

$4 y = - 6 \sqrt{2} + 3$

Divide each term in $4 y = - 6 \sqrt{2} + 3$ by $4$ and simplify.

Divide each term in $4 y = - 6 \sqrt{2} + 3$ by $4$ .

$\frac{4 y}{4} = \frac{- 6 \sqrt{2}}{4} + \frac{3}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y}{4} = \frac{- 6 \sqrt{2}}{4} + \frac{3}{4}$

Divide $y$ by $1$ .

$y = \frac{- 6 \sqrt{2}}{4} + \frac{3}{4}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 6$ and $4$ .

Factor $2$ out of $- 6 \sqrt{2}$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$y = 2.87132034 \dots, - 1.37132034 \dots$

Solve Using the Square Root Property (4y-3)^2=72

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