Two electric dipoles, A, B with respective dipole moments $$\vec{d_A} = -4qa\hat{i}$$ and $$\vec{d_B} = -2qa\hat{i}$$ are placed on the $$x$$-axis with a separation $$R$$, as shown in the figure.

The distance from A at which both of them produce the same potential is:

Let a point lie to the right of B at a distance $$x$$ from A.

$$\text{Distance from A } = x \implies \theta_A = 180^\circ$$

$$\text{Distance from B } = x - R \implies \theta_B = 180^\circ$$

Equating potentials ($$V_A = V_B$$): $$\frac{k(-4qa)}{x^2} = \frac{k(-2qa)}{(x-R)^2}$$

$$\frac{2}{x^2} = \frac{1}{(x-R)^2} \implies \frac{\sqrt{2}}{x} = \frac{1}{x-R}$$

$$\sqrt{2}x - \sqrt{2}R = x \implies x(\sqrt{2}-1) = \sqrt{2}R \implies x = \frac{\sqrt{2}R}{\sqrt{2}-1}$$