We are given that \(x - 3\) is directly proportional to \(y^2\). Mathematically, this can be represented as:

\(x - 3 = ky^2\)

where \(k\) is the constant of proportionality.

We are also given that when \(x = 5\) and \(y = 2\), we have:

\(5 - 3 = k(2^2)\)

\(2 = 4k\)

\(k = \frac{2}{4}\)

\(k = \frac{1}{2}\)

Now that we have the value of \(k\), we can use it to find \(x\) when \(y = 6\):

\(x - 3 = \frac{1}{2}(6^2)\)

\(x - 3 = \frac{1}{2}(36)\)

\(x - 3 = 18\)

\(x = 18 + 3\)

\(x = 21\)

Therefore, when \(y = 6\), \(x\) is equal to 21.