how do you solve for zero for this equation?
(2/3)X -12 -(20/X^2)
(2/3)X -12 -(20/X^2) = 0 -->
2/3 X^3 - 12 X^2 - 20 = 0 --->
X^3 - 18 X^2 - 30 = 0
Numerical solution:
You see that for X = 18 the first and second term cancel, so the left hand side is -30. For X that large this is relatively close to zero (e.g. for X = 10 the left hand side is -830). Having found an approximate root, you can improve it by using Newton's method. If X_0 is your first guess of the solution of the equation f(X) = 0, then a better solutionis:
X_1 = X_0 - f(X_0)/f'(X_0)
In our case:
f(X) = X^3 - 18 X^2 - 30 --->
f'(X) = 3 X^2 - 36X
X_1 = 18.09
Iterating by replacing X_0 by X_1 gives:
X_2 = 18.091657
Iterating again:
X_3 = 18.0916567755
f(X_3) = -1*10^(-8)
so we are already very close to the exact solution.
Exact solution:
X^3 - 18 X^2 - 30 = 0
First we need to get rid of the quadratic term. We substitute:
X = Y + 18/3 = Y + 6
Then
X^3 = Y^3 + 18 Y^2 + 108 Y + 216
X^2 = Y^2 + 12 Y + 36
And we see that in terms of Y the equation is
X^3 - 18 X^2 - 30 =
Y^3 - 108 Y - 462 = 0
An equations of the form:
Y^3 + p Y + q = 0
can be transformed into a quadratic equation by substituting:
Y = Z - p/(3Z)
Y^3 in terms of Z is:
Z^3 - 3 Z^2 p/(3Z) + 3Zp^2/(9Z^2) - p^3/(27 Z^3) =
Z^3 - p Z + p^2/(3Z) - p^3/(27 Z^3)
And we find that:
Y^3 + p Y + q =
Z^3 - p^3/(27 Z^3) + q = 0
Multiplying by Z^3 gives:
Z^6 + q Z^3 - p^3/27 = 0
This is a quadratic equation in Z^3. The solutions are:
Z^3 = -q/2 +/- sqrt[(p/3)^3 + (q/2)^2]
Note that 1/Z^3 can be simplified by multiplying the numerator and denominator by
q/2 +/- sqrt[(p/3)^3 + (q/2)^2]
the result is:
1/Z^3 = (3/p)^3 *
[q/2 +/- sqrt[(p/3)^3 + (q/2)^2]]
And it follows from this that a solution for Y is:
Y = Z - p/(3Z) =
(W - q/2)^1/3 - (W + q/2)^1/3
where
W = sqrt[(p/3)^3 + (q/2)^2]
In general the other three solutyions are found by multiplying Z by a cube root of unity exp[2 pi i n /3].
In our case p = -108 and q = -462. This gives:
W = sqrt[6705] -->
Y = [sqrt(6705) + 231]^1/3 -
[sqrt(6705) - 231]^1/3
and X = Y + 6 =
6 + [sqrt(6705) + 231]^1/3 -
[sqrt(6705) - 231]^1/3
= 18.0916567755
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