Where is the net electric field zero?

[march21]

Homework Statement

Particle 1 of charge +4.0 μC and particle 2 of charge +1.0 μC are held at separation L=10.0 cm on an x axis. Particle 3 of unknown charge q3 is to be located such that the net electrostatic force on it from particles 1 and 2 is zero.
In the included figure, particle 1 is located at the origin and particle 2 is located 10 cm to the left on the x axis.

Homework Equations

E=kq/|r|² * r hat
Enet=E1+E1
F=E*q

The Attempt at a Solution

I thought that if the net electrostatic force is zero, then Enet must be zero. So the two electric fields created by particle 1 and particle 2 must cancel each other other. Following that thought:
E1=q1*k/|r1|² * r1 hat
E2=q1*k/|r2|² * r1 hat

r1 = r_observation - r_q1
r1 = <x,0,0>
|r1|=x
r1 hat= <1,0,0>

r2=r_observation - r_q2
r2=<x-0.10,0,0> m
|r2|=x-0.10
r2 hat = <1,0,0>

Enet=E1+E2
0=E1+E2
E1=-E2
k*q1*r1 hat/x² = k*q2*r2hat/(x²-0.20x+0.01)
After simplifying:
x=[(-q2/q1)-0.01]/-0.20
x=1.3 m

However this answer does not solve the problem or make sense. I was thinking that in order for the two electric fields to cancel each other out, the third particle should be placed in between the two particles, closer to the more weakly charged. I've played around with these equations several times and still can't figure it out. Thank you in advance for any help, and I apologize for any formatting errors.

 

[Doc Al]

I was thinking that in order for the two electric fields to cancel each other out, the third particle should be placed in between the two particles, closer to the more weakly charged.

Of course.

I think you're messing things up by trying to put the coordinates in the equations. Instead, let x be the distance from particle 2 and thus .1 - x is the distance from particle 1. Set up a simple equation so that the field magnitudes are equal and solve for x. (You can always translate that to the coordinate later.)

 

[march21]

Ohhhh, that makes much more sense.
Thank you!