Answer
$x=\left\{ -\dfrac{3}{5},4 \right\}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
5x^2-17x=12
,$ 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}
5x^2-17x-12=0
.\end{array}
Using factoring of trinomials, the value of $ac$ in the trinomial expression above is $
5(-12)=-60
$ and the value of $b$ is $
-17
.$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{
3,-20
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
5x^2+3x-20x-12=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}
(5x^2+3x)-(20x+12)=0
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
x(5x+3)-4(5x+3)=0
.\end{array}
Factoring the $GCF=
(5x+3)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(5x+3)(x-4)=0
.\end{array}
Equating each factor to zero (Zero Product Property), the solutions to the equation above are
\begin{array}{l}\require{cancel}
5x+3=0
\\\\\text{OR}\\\\
x-4=0
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
5x+3=0
\\\\
5x=-3
\\\\
x=-\dfrac{3}{5}
\\\\\text{OR}\\\\
x-4=0
\\\\
x=4
.\end{array}
Hence, $
x=\left\{ -\dfrac{3}{5},4 \right\}
.$
