Given \(x = 3 - \sqrt{3}\), we want to find \(x^2 + \frac{36}{x^2}\).

Let's start by calculating \(x^2\):

\[x^2 = (3 - \sqrt{3})^2\]

\[x^2 = 9 - 6\sqrt{3} + 3\]

\[x^2 = 12 - 6\sqrt{3}\]

Now, let's calculate \(\frac{36}{x^2}\):

\[\frac{36}{x^2} = \frac{36}{12 - 6\sqrt{3}}\]

To simplify the denominator, we can rationalize it by multiplying both the numerator and denominator by the conjugate of the denominator:

\[\frac{36}{x^2} = \frac{36 \cdot (12 + 6\sqrt{3})}{(12 - 6\sqrt{3}) \cdot (12 + 6\sqrt{3})}\]

\[\frac{36}{x^2} = \frac{432 + 216\sqrt{3}}{144 - (6\sqrt{3})^2}\]

\[\frac{36}{x^2} = \frac{432 + 216\sqrt{3}}{144 - 108}\]

\[\frac{36}{x^2} = \frac{432 + 216\sqrt{3}}{36}\]

\[\frac{36}{x^2} = 12 + 6\sqrt{3}\]

Now, let's add \(x^2\) and \(\frac{36}{x^2}\):

\[x^2 + \frac{36}{x^2} = (12 - 6\sqrt{3}) + (12 + 6\sqrt{3})\]

\[x^2 + \frac{36}{x^2} = 24\]

Therefore, \(x^2 + \frac{36}{x^2} = 24\).