Solve Using the Square Root Property (x-1/2)^2=121/4

${(x - \frac{1}{2})}^{2} = \frac{121}{4}$

Step 1

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

$x - \frac{1}{2} = \pm \sqrt{\frac{121}{4}}$

Step 2

Simplify $\pm \sqrt{\frac{121}{4}}$ .

Rewrite $\sqrt{\frac{121}{4}}$ as $\frac{\sqrt{121}}{\sqrt{4}}$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{121}}{\sqrt{4}}$

Simplify the numerator.

Rewrite $121$ as $11^{2}$ .

$x - \frac{1}{2} = \pm \frac{\sqrt{11^{2}}}{\sqrt{4}}$

Pull terms out from under the radical, assuming positive real numbers.

$x - \frac{1}{2} = \pm \frac{11}{\sqrt{4}}$

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$x - \frac{1}{2} = \pm \frac{11}{\sqrt{2^{2}}}$

Pull terms out from under the radical, assuming positive real numbers.

$x - \frac{1}{2} = \pm \frac{11}{2}$

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.

$x - \frac{1}{2} = \frac{11}{2}$

Move all terms not containing $x$ to the right side of the equation.

Add $\frac{1}{2}$ to both sides of the equation.

$x = \frac{11}{2} + \frac{1}{2}$

Combine the numerators over the common denominator.

$x = \frac{11 + 1}{2}$

Add $11$ and $1$ .

$x = \frac{12}{2}$

Divide $12$ by $2$ .

$x = 6$

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

$x - \frac{1}{2} = - \frac{11}{2}$

Move all terms not containing $x$ to the right side of the equation.

Add $\frac{1}{2}$ to both sides of the equation.

$x = - \frac{11}{2} + \frac{1}{2}$

Combine the numerators over the common denominator.

$x = \frac{- 11 + 1}{2}$

Add $- 11$ and $1$ .

$x = \frac{- 10}{2}$

Divide $- 10$ by $2$ .

$x = - 5$

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

$x = 6, - 5$