Answer
2 rational solutions
Work Step by Step
$\bf{\text{Solution Outline:}}$
To evaluate the discriminant of the given equation, $
8x^2-72=0
,$ identify first the value of $a,b,$ and $c$ in the quadratic expression $ax^2+bx+c.$ Then use the Discriminant Formula. If the value of the discriminant is less than zero, then there are $\text{
2 nonreal complex solutions
.}$ If the value is $0,$ then there is $\text{
1 distinct rational solution
.}$ If the value of the discriminant is a positive perfect square, then there are $\text{
2 rational solutions
.}$ Finally, if the value of the discriminant is positive but not a perfect square, there are $\text{
2 irrational solutions
.}$
$\bf{\text{Solution Details:}}$
In the equation above, $a=
8
,$ $b=
0
,$ and $c=
-72
.$ Using the Discriminant Formula which is given by $b^2-4ac,$ the value of the discriminant is
\begin{array}{l}\require{cancel}
(0)^2-4(8)(-72)
\\\\=
2304
\\\\=
48^2
.\end{array}
Since the discriminant is $\text{
a positive perfect square
,}$ then there are $\text{
2 rational solutions
.}$