Try to explain what is $\Theta$ Notation.

$\frac{1}{2}x^{2} - 3n = \Theta(n^{2})$

What $\Theta$ Notation says LHS and RHS are almost equal Not neccesorily exact equal ( by some constant).

$c_{1}x^{2} \leq \frac{1}{2}x^{2} - 3n \leq c_{2}n^{2}$

Next he cancle n^{2}

$c_{1} \leq \frac{1}{2} - \frac{3}{n} \leq c_{2}$

Now we can take some values of n then Both LHS and RHS are same.