Q. 3 - Intermediate Algebra

Question

Solve Quadratic Equations of the Form $a{x}^2=k$ Using the Square Root Property.

In the following exercises, solve each equation  $r^2-24=0$

Step-by-Step Solution

Answer

The solution of the quadratic equation:  $r=2\sqrt{6}, r=-2\sqrt{6}$

Step 1. Given information

Consider the quadratic equation $r^2-24=0$

The objective is to solve the quadratic equation by using the square root property. 

If $x^2=k$, then $x=\pm \sqrt{k}$.

Isolate the quadratic term and make its coefficient one. Add 24 to both sides

$r^2=24$

Step 2. Apply square root property

So, $r=\pm \sqrt{24}$

Simplify the radical.   

$$\begin{gathered} r=\pm \sqrt{4·6} \\ r=\pm 2\sqrt{6} \end{gathered}$$

Rewrite to show two solutions.  

$r=2\sqrt{6}, r=-2\sqrt{6}$

Step 3. Check the solutions

Substitute $r=2\sqrt{6}$ in the original equation.

$$\begin{gathered} r^2-24=0 \\ {\left(2\sqrt{6}\right)}^2-24\stackrel{?}{=}0 \\ 24-24\stackrel{?}{=}0 \\ 0=0 \end{gathered}$$

Substitute $r=-2\sqrt{6}$in the original equation.

$$\begin{gathered} r^2-24=0 \\ {(-2\sqrt{6})}^2-24\stackrel{?}{=}0 \\ 24-24\stackrel{?}{=}0 \\ 0=0 \end{gathered}$$