Find the utility maximizing bundle [Sundaram, P.169, Q.7 (Kuhn-Tucker Theorem) ]
The equation $100\sqrt{x_1} = 8x_1^{3/2} - 5$ is cubic in $\sqrt{x_1}$.  Unless you have been asked for an analytic solution (which is possible but complex), the approximate method I suggest is first to ignore the minus $5$, enabling division by $\sqrt{x_1}$ yielding:
$$100 = 8x_1$$
The first approximation is then $x_1 = 100/8 = 12.5$. Since $100\sqrt{12.5} \approx 354$ is much greater than $5$, it can be inferred that the exact solution will be very close to $12.5$.  So try successive approximation, perhaps starting with $x_1=12.6$.