Use the quadratic formula to solve for $x$
$$6x^2-9x+1=0$$

Step 1 ：Given the quadratic equation $6x^2-9x+1=0$, we are asked to find the solutions for $x$.

Step 2 ：We can use the quadratic formula, which is given by $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, where $a$, $b$, and $c$ are coefficients of the quadratic equation $a{x}^2+bx+c=0$.

Step 3 ：In this case, $a=6$, $b=-9$, and $c=1$. We can substitute these values into the quadratic formula to find the solutions for $x$.

Step 4 ：First, calculate the discriminant $D$, which is $b^2-4ac$. Substituting the given values, we get $D=57$.

Step 5 ：Next, substitute $a$, $b$, and $D$ into the quadratic formula to find the solutions for $x$. The solutions are $x1=1.379152869605896$ and $x2=0.12084713039410418$.

Step 6 ：Finally, rounding to three decimal places, the solutions to the equation $6x^2-9x+1=0$ are $x=\boxed{{\displaystyle 1.379}}$ and $x=\boxed{{\displaystyle 0.121}}$.