Q. 27 - Intermediate Algebra

Question

In the following exercises, solve each equation.

${\left(r-\frac{1}{2}\right)}^2=\frac{3}{4}$

Step-by-Step Solution

Answer

The solution of the equation is $r=\frac{1+\sqrt{3}}{2},\frac{1-\sqrt{3}}{2}$

Step 1. Given information

In the given question we have to solve given equation. 

${\left(r-\frac{1}{2}\right)}^2=\frac{3}{4}$

Step 2. The given equation is r − 1 2 2 = 3 4

Use the Square Root Property.

$r-\frac{1}{2}=\pm \sqrt{\frac{3}{4}}$

Step 3. Simplify the radical.

$$\begin{gathered} r-\frac{1}{2}=\pm \frac{\sqrt{3}}{\sqrt{4}} \\ r-\frac{1}{2}=\pm \frac{\sqrt{3}}{2} \end{gathered}$$

Add $\frac{1}{2}$on both side

$$\begin{gathered} r-\frac{1}{2}+\frac{1}{2}=\frac{1}{2}\pm \frac{\sqrt{3}}{2} \\ r=\frac{1}{2}\pm \frac{\sqrt{3}}{2} \end{gathered}$$

Step 4. Solve for r

$$\begin{gathered} r=\frac{1}{2}+\frac{\sqrt{3}}{2} r=\frac{1}{2}-\frac{\sqrt{3}}{2} \\ r=\frac{1+\sqrt{3}}{2} r=\frac{1-\sqrt{3}}{2} \end{gathered}$$