Optimizing Electrostatics Question: Minimizing Force on Particle 3

[RoyalFlush100]

Homework Statement

In the figure particles 1 and 2 are fixed in place on an x axis, at a separation of L = 7.30 cm. Their charges are q1 = +e and q2 = -27e. Particle 3 with charge q3 = +3e is to be placed on the line between particles 1 and 2, so that they produce a net electrostatic force on it.
At what coordinate should particle 3 be placed to minimize the magnitude of that force?

The figure shows particle 1 at x=0 and particle 2 at x=7.30 cm.

Homework Equations

F=kqq/r^2
Setting x equal to the x coordinate where q3 would be.

The Attempt at a Solution

F = k3e^2(x^-2+[27]/[L^2 - 2Lx + x^2]) (factored out k, q3 [3e], and then e from the two charges)
So from here you want to find the critical points. So I took the derivative and set it equal to 0 (also got rid of 3ke^2):
0 = -2x^-3 + (0 - 27[-2L+2x])/(L^2 - 2Lx + x^2)^2
I rearranged it to this (assuming I made no mistakes):
0 = x^-3 * [56x^4 - (62L)x^3 + (12L^2)x^2 - (8L^3)x + 2L^4]
I have no idea how to solve for x in this case and we're not permitted graphing calculators, so how would you attempt this question?

 

[TSny]

I recommend not squaring out the ##(L-x)^2## in the denominator of the second force. Thus, you want to minimize the function ##f(x) = \frac{1}{x^2} + \frac{27}{(L-x)^2}##.

 

[RoyalFlush100]

Thanks, now I got it.