Answer
$x=\left\{ -\dfrac{3}{2},\dfrac{1}{3} \right\}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given eqution, $
6x^2+7x=3
,$ express the equation in the form $ax^2+bx+c=0.$ Then express the equation in factored form. Next step is to equate each factor to zero (Zero Product Property). Finally, solve each equation.
$\bf{\text{Solution Details:}}$
Using the properties of equality, the equation above is equivalent to
\begin{array}{l}\require{cancel}
6x^2+7x-3=0
.\end{array}
Using factoring of trinomials, the value of $ac$ in the trinomial expression above is $
6(-3)=-18
$ and the value of $b$ is $
7
.$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{
-2,9
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
6x^2-2x+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}
(6x^2-2x)+(9x-3)=0
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
2x(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)(2x+3)=0
.\end{array}
Equating each factor to zero (Zero Product Property), the solutions to the equation above are
\begin{array}{l}\require{cancel}
3x-1=0
\\\\\text{OR}\\\\
2x+3=0
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
3x-1=0
\\\\
3x=1
\\\\
x=\dfrac{1}{3}
\\\\\text{OR}\\\\
2x+3=0
\\\\
2x=-3
\\\\
x=-\dfrac{3}{2}
.\end{array}
Hence, $
x=\left\{ -\dfrac{3}{2},\dfrac{1}{3} \right\}
.$
