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There are a number of ways. One possibility is the iterative method.
Rewrite the equation so as to make x the subject:
4x3 + 0.4x2 + 0.01x - 2.4*10-18 = 0
4x3 = 2.4*10-18 - 0.4x2 - 0.01x
x3 = (2.4*10-18 - 0.4x2 - 0.01x)/4 x = cuberoot[(2.4*10-18 - 0.4x2 - 0.01x)/4]
The original equation is approximately equal to
4x3 + 0.4x2 + 0.01x = 0
or x*(20x - 1)2 = 0
which has roots at x = 0 and x = -0.05
Start with a value, x1 near one of these roots and use the iteration:
xn+1 = cuberoot[(2.4*10-18 - 0.4xn2 - 0.01xn)/4] where n = 1, 2, 3, etc to improve your estimates.
The resulting roots are, not surprisingly, 0 and -0.05 to 10 decimal places.
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