Coulomb's law:

$\overline{){\mathbf{F}}{\mathbf{=}}\frac{\mathbf{k}{\mathbf{q}}_{\mathbf{1}}{\mathbf{q}}_{\mathbf{2}}}{{\mathbf{r}}^{\mathbf{2}}}}$

Applying Coulomb's law between 3q and q_{0}:

$\begin{array}{rcl}\mathbf{F}& {}_{\mathbf{1}}\mathbf{=}& \mathbf{k}\frac{\mathbf{\left(}\mathbf{3}\mathbf{q}\mathbf{\right)}\mathbf{\left(}{\mathbf{q}}_{\mathbf{0}}\mathbf{\right)}}{{\mathbf{x}}^{\mathbf{2}}}\\ & \mathbf{=}& \mathbf{k}\frac{\mathbf{3}\mathbf{q}{\mathbf{q}}_{\mathbf{0}}}{{\mathbf{x}}^{\mathbf{2}}}\end{array}$

Applying Coulomb's law between -q and q_{0}:

$\begin{array}{rcl}\mathbf{F}& {}_{\mathbf{2}}\mathbf{=}& \mathbf{k}\frac{\mathbf{(}\mathbf{-}\mathbf{q}\mathbf{)}\mathbf{\left(}{\mathbf{q}}_{\mathbf{0}}\mathbf{\right)}}{{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{D}\mathbf{)}}^{\mathbf{2}}}\\ & \mathbf{=}& \mathbf{-}\mathbf{k}\frac{{\mathbf{qq}}_{\mathbf{0}}}{{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{5}\mathbf{)}}^{\mathbf{2}}}\end{array}$

A point charge +3q is located at the origin, and a point charge -q is located on the x-axis at d = 0.5 m. At what location on the x-axis will a third charge q_{0} experience no net force from the other two charges?

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