Answer
2 rational solutions
Work Step by Step
$\bf{\text{Solution Outline:}}$
To evaluate the discriminant of the given equation, $
3x^2+5x+2=0
,$ identify first the value of $a,b,$ and $c.$ 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, $a=
3
,$ $b=
5
,$ and $c=
2
.$ Using the Discriminant Formula which is given by $b^2-4ac,$ the value of the discriminant is
\begin{array}{l}\require{cancel}
(5)^2-4(3)(2)
\\\\=
25-24
\\\\=
1
.\end{array}
Since the discriminant is $\text{
a positive perfect square
,}$ then there are $\text{
2 rational solutions
.}$