Answer
$\left\{\dfrac{1}{5},2\right\}$
Work Step by Step
In the form $ax^2+bx+c=0,$ the given equation, $
5x^2=11x-2
,$ is equivalent to
\begin{align*}
5x^2-11x+2=0
.\end{align*}
Using the factoring of trinomials in the form $ax^2+bx+c,$ the equation above has $ac=
5(2)=10
$ and $b=
-11
.$
The two numbers with a product of $c$ and a sum of $b$ are $\left\{
-1,-10
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
5x^2-x-10x+2=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-x)-(10x-2)=0
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
x(5x-1)-2(5x-1)=0
.\end{array}
Factoring the $GCF=
(5x-1)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(5x-1)(x-2)=0
.\end{array}
Equating each factor to zero (Zero Product Property) and solving for the variable, then
\begin{array}{l|r}
5x-1=0 & x-2=0
\\
5x=1 & x=2
\\\\
x=\dfrac{1}{5}
\end{array}
Hence, the solution set of the equation $
5x^2=11x-2
$ is $\left\{\dfrac{1}{5},2\right\}$.
