Answer
$x=\dfrac{1}{3}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use the properties of equality and express the given equation, $
9x^2=6x-1
,$ in the form $ax^2+bx+c=0.$ Then express the equation in factored form. Once in factored form, equate each factor to zero (Zero Product Property). Finally, solve each resulting equation.
$\bf{\text{Solution Details:}}$
Using the properties of equality, the equation above is equivalent to
\begin{array}{l}\require{cancel}
9x^2-6x+1=0
.\end{array}
Using factoring of trinomials, the value of $ac$ in the trinomial expression above is $
9(1)=9
$ and the value of $b$ is $
-6
.$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{
-3,-3
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
9x^2-3x -3x +1 =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}
(9x^2-3x )-(3x -1)=0
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
3x(3x-1 )-(3x -1)=0
.\end{array}
Factoring the $GCF=
(3x-1 )
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(3x-1 )(3x-1)=0
\\\\
(3x-1 )^2=0
.\end{array}
Getting the square root of both sides (Square Root Principle), the equation above is equivalent to
\begin{array}{l}\require{cancel}
3x-1=\pm\sqrt{0}
\\\\
3x-1=0
\\\\
3x=1
\\\\
x=\dfrac{1}{3}
.\end{array}
Hence, the solution is $
x=\dfrac{1}{3}
.$
