Two point charges ( +Q) and (-2Q) are fixed on the x-axis at positions a and 2a from origin, respectively. At what positions on the axis, the resultant electric field is zero?

Suppose electric field is zero at a point P which lies at a distanced from the charge+ Q.

$$\begin{gathered} \text{ At }\mathrm{P}\frac{\mathrm{kQ}}{{\mathrm{d}}^2}=\frac{\mathrm{k}\left(2\mathrm{Q}\right)}{(\mathrm{a}+\mathrm{d}{)}^2} \\ \Rightarrow \frac{{\displaystyle 1}}{{\displaystyle {\mathrm{d}}^2}}=\frac{{\displaystyle 2}}{{\displaystyle (\mathrm{a}+\mathrm{d}{)}^2}}\Rightarrow \mathrm{d}=\frac{{\displaystyle \mathrm{a}}}{{\displaystyle (\sqrt{2}-1)}} \end{gathered}$$

Since d > a, i.e., point P must lies on negative x-axis as shown at a distance x from origin. Hence

$\mathrm{x}=\mathrm{d}-\mathrm{a}=\frac{\mathrm{a}}{(\sqrt{2}-1)}-\mathrm{a}=\sqrt{2}\mathrm{a}$

Actually P lies on negative x-axis, so $x=-\sqrt{2}a$