#### Answer

$\text{Discriminant: }
76
\\\text{Number of distinct solutions: }
2
\\\text{Type of solutions:
irrational solutions
}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To evaluate the discriminant of the given equation, $
-6x^2+2x=-3
,$ identify first the values 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 $\text{
1 rational solution
.}$ 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 form $ax^2+bx+c=0,$ the given equation is equivalent to
\begin{array}{l}\require{cancel}
-6x^2+2x+3=0
\\\\
-1(-6x^2+2x+3)=-1(0)
\\\\
6x^2-2x-3=0
.\end{array}
In the equation above, $a=
6
,$ $b=
-2
,$ and $c=
-3
.$ Using the Discriminant Formula which is given by $b^2-4ac,$ the value of the discriminant is
\begin{array}{l}\require{cancel}
(-2)^2-4(6)(-3)
\\\\=
4+72
\\\\=
76
.\end{array}
Since the discriminant is $\text{
is greater than zero and not a perfect square
,}$ then there are $\text{
2 irrational solutions
.}$
Hence, the given equation has the following properties:
\begin{array}{l}\require{cancel}
\text{Discriminant: }
76
\\\text{Number of distinct solutions: }
2
\\\text{Type of solutions:
irrational solutions
}
.\end{array}