Q.6 - 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 $v^2-80=0$

Step-by-Step Solution

Answer

The solution of the quadratic equation: $v=4\sqrt{5}, v=-4\sqrt{5}$

Step 1: Given information

Consider the quadratic equation  $v^2-80=0$

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 80 to both sides.

$v^2=80$

Step 2. Apply square root property

So, $v=\pm \sqrt{80}$

Simplify the radical.  

$$\begin{gathered} v=\pm \sqrt{16·5} \\ v=\pm 4\sqrt{5} \end{gathered}$$

Rewrite to show two solutions.   

$v=4\sqrt{5}, v=-4\sqrt{5}$

Step 3. Check the solutions

Substitute $v=4\sqrt{5}$ in the original equation.

$$\begin{gathered} v^2-80=0 \\ {\left(4\sqrt{5}\right)}^2-80\stackrel{?}{=}0 \\ 80-80\stackrel{?}{=}0 \\ 0=0 \end{gathered}$$

Substitute $v=-4\sqrt{5}$in the original equation.

$$\begin{gathered} v^2-80=0 \\ {(-4\sqrt{5})}^2-80\stackrel{?}{=}0 \\ 80-80\stackrel{?}{=}0 \\ 0=0 \end{gathered}$$