Answer
$\left\{\dfrac{1}{2},4\right\}$
Work Step by Step
In the form $ax^2+bx+c=0,$ the given equation, $
2x^2=9x-4
,$ is equivalent to
\begin{align*}
2x^2-9x+4=0
.\end{align*}
Using the factoring of trinomials in the form $ax^2+bx+c,$ the equation above has $ac=
2(4)=8
$ and $b=
-9
.$
The two numbers with a product of $c$ and a sum of $b$ are $\left\{
-1,-8
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
2x^2-x-8x+4=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}
(2x^2-x)-(8x-4)=0
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
x(2x-1)-4(2x-1)=0
.\end{array}
Factoring the $GCF=
(2x-1)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(2x-1)(x-4)=0
.\end{array}
Equating each factor to zero (Zero Product Property) and solving for the variable, then
\begin{array}{l|r}
2x-1=0 & x-4=0
\\
2x=1 & x=4
\\\\
x=\dfrac{1}{2}
\end{array}
Hence, the solution set of the equation $
2x^2=9x-4
$ is $\left\{\dfrac{1}{2},4\right\}$.
