(b) Suppose in the following figure, equilibrium of charge $B$ is considered. Hence for it's equilibrium $|{F_A}|\, = \,|{F_C}|$
$==>$ $\frac{1}{{4\pi {\varepsilon _0}}}\frac{{{Q^2}}}{{4{x^2}}} = $$\frac{1}{{4\pi {\varepsilon _0}}}\frac{{qQ}}{{{x^2}}}$ $==>$ $q = \frac{{ – \,Q}}{4}$

Short Trick : For such type of problem the magnitude of middle charge can be determined if either of the extreme charge is in equilibrium by using the following formula.
If charge $A$ is in equilibrium then $q$ $= -$ ${Q_B}\,{\left( {\frac{{{x_1}}}{x}} \right)^2}$
If charge $B $ is in equilibrium then $q = – {Q_A}\,\,{\left( {\frac{{{x_2}}}{x}} \right)^2}$
If the whole system is in equilibrium then use either of the above formula.