Three Charges Aligned on a Straight Wire

[Thundies22]

Three charges are aligned along a straight wire as show.
+C ?<--- -3/2 C --->? -C
+ q₁ -3/2 q₃ - q₂
X----------------------------o-----------------------------X
P(midpoint)
|---------------------------- L-----------------------------2L
x=0 -----> +x

Assume that the +q₁ charge is situated at the origin and the -q₂ charge is located at distance 2L from the origin. Determine the distance that the -1.5q₃ charge is from the origin in order that the total electric field at the midpoint of the wire vanishes.

Of course we have the positive charge (E field radially outward) and negative charge (E field radially inward) both directing in the +ve x direction. So we would expect to see the charge -3/2q₃ to the left of P (-x) to cancel the two fields in the +x direction from q₁ and q₂. We would use E_total = 1/4πε_o { E1 + E2- E3 } = 0 and find the distance x by finding a root. Where E₁= q/(0-L)², E₂= q/(2L-L)², E₃=3q/2(x-L)².

I tried this one a few times just going in loops and don't want to confuse anyone one else. For some reason this question which is a trivial 1D problem is getting me lol. Thanks for your support.

 

[haruspex]

Looks like your diagram has been messed up by the formatting. To avoid that, enclose such ASCII drawings in

 ...

. (Click Reply to see what I wrote.)
Also confused by your charge notation. Do you mean charge q₁=-1.5q, etc?

 

[Delta2]

You already have setup the equation correctly (if i understand correctly your three charges are q,-q and -1.5q).

What prevents you from solving the equation for x? You will find two solutions for x is that what confuses you?