How to Find R from Electric Charge Problem

[TSny]

I am getting confused.. wasn't F_x i shoud have used..

Exactly, F_x is what you should use. The force that q₃ exerts on q₁ is

F = kq₁q₃/r²

The x-component of this force is F_x = Fcosθ. So,

F_x = kq₁q₃cosθ/r².

q₄ creates the same x-component of force on q₁. So, the total force on q₁ from q₃ and q₄ together is

2kq₁q₃cosθ/r² $\;\;\;\;$ (in the positive x direction)

q₂ creates a force on q₁ in the negative x direction of magnitude kq₁q₂/R².

So, the net force will be zero if

2kq₁q₃cosθ/r² = kq₁q₂/R²

This leads to

2q₃cosθ/r² = q₂/R²

And the error, I really don't understand how there can be a error in the equation i wrote..

Right. That's what we need to determine. So, let's go through the steps for solving the last equation above for R. I suggested what I thought was a good first step; namely, to multiply the equation through by the least common denominator r²R². If that's not how you want to do it, that's ok. But, can you please show your next step or two in solving for R?