Answer
$$\text{A}$$
Work Step by Step
Recall:
A quadratic equation in the form $ax^2+bx+c=0$ can be solved using the Quadratic Formula:
$$x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}$$
The given quadratic equation has $a=6, b=-10,$ and $c=3$.
Use these values and the quadratic formula to obtain:
\begin{align*}
x&=\dfrac{-(-10)\pm\sqrt{(-10)^2-4(6)(3)}}{2(6)}\\\\
&=\dfrac{10\pm \sqrt{100-72}}{12}\\\\
&=\dfrac{10\pm \sqrt{28}}{12}\\\\
&=\dfrac{10\sqrt{4(7)}}{12}\\\\
&=\dfrac{10\pm 2\sqrt7}{12}
\end{align*}
Cancel out the common factor of $2$ to obtain:
\begin{align*}
\require{cancel}
x&=\dfrac{\cancel{10}^5\pm \cancel{2}\sqrt7}{\cancel{12}^6}\\
&=\dfrac{5\pm \sqrt7}{6}
\end{align*}
Thus, the answer is Option $\text{A}$.
