Solve for x in this question~~

From the equation, it seems difficult to solve it by analytical method. Instead, we should use numerical method to approximate the value of x.

Let f(x) = x^3/2 + 12x - 1

f'(x) = 3/2 x^1/2 + 12

By Newton's method,

The root of f(x) = 0 can be approximated by the iteration equation

xn+1 = xn - f(xn)/f'(xn)

With the initial guess x0 = 0, f(x0) = -1

x1 = x0 - f(x0)/f'(x0) = 0.083333333

x2 = x1 - f(x1)/f'(x1) = 0.081398463

x3 = x2 - f(x2)/f'(x2) = 0.08139807


So, corrected to 4 d.p., the root of f(x) = 0 is x = 0.0814