Answer
$\left\{-3,\dfrac{1}{3}\right\}$
Work Step by Step
Using the factoring of trinomials in the form $ax^2+bx+c,$ the $\text{
equation
}$
\begin{array}{l}\require{cancel}
3x^2+8x-3=0
\end{array} has $ac=
3(-3)=-9
$ and $b=
8
.$
The two numbers with a product of $c$ and a sum of $b$ are $\left\{
-1,9
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
3x^2-x+9x-3=0
.\end{array}
Grouping the first and second terms and the third and fourth terms, the given expression is equivalent to
\begin{array}{l}\require{cancel}
(3x^2-x)+(9x-3)=0
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
x(3x-1)+3(3x-1)=0
.\end{array}
Factoring the $GCF=
(3x-1)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(3x-1)(x+3)=0
.\end{array}
Equating each factor to zero (Zero Product Property) and solving for the variable, then
\begin{array}{l|r}
3x-1=0 & x+3=0
\\
3x=1 & x=-3
\\\\
x=\dfrac{1}{3}
\end{array}
Hence, the solution set of the equation $
3x^2+8x-3=0
$ is $\left\{-3,\dfrac{1}{3}\right\}$.
