To solve the equation \(4.8^{2} + 28.6^{2} = x^{2}\), we start by calculating \(4.8^{2}\) and \(28.6^{2}\).
Calculating \(4.8^{2}\):
\[ 4.8^{2} = 23.04 \]
Calculating \(28.6^{2}\):
\[ 28.6^{2} = 817.96 \]
Now, we add these two results together:
\[ 4.8^{2} + 28.6^{2} = 23.04 + 817.96 = 841 \]
Thus, we have:
\[ x^{2} = 841 \]
Next, we find \(x\) by taking the square root of both sides:
\[ x = \sqrt{841} = 29 \]
Therefore, the solution is
\[ \boxed{29} \]