Find the two possible values x3,r and x3,ℓ for the position of charge 3.

Given:

q1 = q
q2 = 4q
q3 = q
d = 2.00 cm
xq1 = 0 cm
xq2 = 2.00 cm
q = 2.00 nC

Let us determine the forces:
F1on3 = K ⋅ |q1| ⋅ |q3| / r2 = K ⋅ |q| ⋅ |q| / (d1-3)2 = K ⋅ q2 / (d1-3)2
F2on3 = K ⋅ |q2| ⋅ |q3| / r2 = K ⋅ |4q| ⋅ |q| / (d3-2)2 = K ⋅ 4q2 / (d3-2)2

We're able to solve for the variables in both cases since the forces are supposed to equal each other in each case.

F1on3 = F2on3
K ⋅ q2 / x2 = K ⋅ 4q2 / (2.00 cm - x)2
1/x2 = 4/ [4.00 cm2 - (4.00 cm⋅x) + x2]
[4.00 cm2 - (4.00 cm ⋅ x) + x2] / x2 = 4
4.00 cm2 / x2 - (4.00 cm ⋅ x) / x2 + x2 / x2 - 4 = 0
4.00 cm2 / x2 - (4.00 cm) / x + 1 - 4 = 0
4.00 cm2 / x2 - (4.00 cm) /x - 3 = 0
3 = 4.00 cm2 / x2 - (4.00 cm) / x
3x2 = 4.00 cm - 4.00 cm ⋅ x
3x2 + 4.00 cm ⋅ x - 4.00 cm = 0

After we plug that into the quadratic equation, we get:
x = -2 & x = 2/3 
x3,r, x3,ℓ = 0.667,-2.00 cm, cm

that is the sum

step-by-step explanation:

there are no other answers without knowing the variables values