The diagram shows the median AU at point A cutting the triangle in half. The length AU is in two parts, $y$ and $z$. Since the triangle is equilateral, $y+z=\frac{\sqrt 3}{2}x$. (You may like to prove this e.g. by trigonometry) To work out $y$ use Pythagoras's Theorem:

\begin{align}

y^2+\left(\frac{1}{2}\right)^2 &=\left(\sqrt 7\right)^2 \\

y^2 &= \frac{27}{4} \\

y &= \frac{3\sqrt 3}{2} 

\end{align}

and use $y+z=\frac{\sqrt 3}{2}x$ to give $z = (x-3)\frac{\sqrt 3}{2}$.

Then more Pythagoras's Theorem and our values found above give

\begin{align}

z^2+\left(\frac{x}{2} - \frac{1}{2}\right)^2 &=2^2 \\

\frac{3(x-3)^2}{4} + \frac{(x-1)^2}{4} = 4

\end{align}

which simplifies to get the same equation as Thom and the others: $x^2 -5x + 3 = 0$.

What a simple solution, Well done!