Find the utility maximizing bundle [Sundaram, P.169, Q.7 (Kuhn-Tucker Theorem) ] A consumer with a utility function given by $u(x_1, x_2) = \sqrt{x_1} + x_1x_2$ has an income of $100$. The unit prices of $x_1$ and $x_2$ are $4$ and $5$, respectively.
(a) Compute the utility-maximizing commodity bundle, if consumption must be nonnegative.
The necessary condition for this optimization is that
$$\partial L/\partial x_1 = \frac1{2\sqrt{x_1}} + x_2 -4\lambda_1+\lambda_2 = 0$$
$$x_1 - 5\lambda_1 + \lambda_3 = 0$$
$$100 - 4x_1 - 5x_2 = 0$$
$$x_1 \ge 0, \quad x_2 \ge 0$$
There are two cases we need to consider $\{x_1 &gt;0, x_2 &gt;0\}$ and $\{x_1 &gt;0, x_2 =0\}$. For the first case, if I solve the above system of equations, I have
$$100\sqrt{x_1} = 8x_1^{3/2} - 5.$$
In this case, how can we obtain the bundle $(x_1^*, x_2^*)$?
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 $x_1=12.6$.