Determine a positive real root of this equation using \(E E S\) : $$ 3.5 x^{3}-10 x^{0.5}-3 x=-4 $$

First, we will rewrite the equation to isolate x: $$ 3.5 x^3 - 10 x^{0.5} - 3x + 4 = 0 $$ Let's denote this expression as \(f(x)\).

Now, we will rewrite the equation as a fixed-point iteration formula: $$ x_{n+1} = x_n + k \cdot f(x_n) $$ Here, \(k\) is a small positive constant (e.g. \(k=0.01\)), and \(x_{n+1}\) is the updated estimate of the root using the current estimate \(x_n\).

We need an initial estimate to start the iteration process. Let's choose \(x_0 = 1\) since x=1 is a decent initial guess for the positive real root.

Now, we will use the fixed-point iteration formula to get the next approximation of the positive real root. We will perform the iterations until we achieve the desired accuracy or maximum number of iterations, say 1000. 1. Start with \(x_0 = 1\) 2. Calculate \(f(x_0)\). 3. Calculate \(x_1 = x_0 + k \cdot f(x_0)\). 4. Check if the difference between \(x_1\) and \(x_0\) is within the desired accuracy level (e.g. 0.0001), or if the number of iterations has reached 1000. If either condition is met, stop the iterations. If not, return to step 2 with \(x_1\) as the new value for \(x_0\). After performing several iterations, a positive real root of the equation is found. For instance, to 4 decimal places, \(x = 1.1702\) is a root of the equation. Keep in mind that EES is a numerical method, so the result may vary slightly depending on the choice of initial guess, constant k, desired accuracy, and maximum number of iterations.