Solving an equation arising from method of image charges

Solving an equation arising from method of image charges

I'm currently studying electrodynamics where the following equation arose:
$0 = (\frac{q}{\sqrt{R^2 + d^2 - 2Rdcos\theta}} + \frac{q_p}{\sqrt{R^2 +
d_p^2 - 2Rd_p cos\theta}})$
where I need to solve for $q_p$ and $d_p$. The solution given is the
following (factoring out $R^2$ out of the squareroot):
$0 = (\frac{q/R}{\sqrt{1 + (d/R)^2 - 2 (d/R) cos\theta}} +
\frac{q_p/d}{\sqrt{1 + (R/d_p)^2 - 2(R/d_p)cos\theta}}) $ which lets us
read out the solutions: $q/R = - q_p/d_p$ and $d/R = R/d_p$ which can be
easily solved for $d_p$ and $q_p$.
My question is the following: Why can't I solve the equation starting from
the first equation? The first equation is zero if we chose $q_p = -q$ and
$d_p = d$. That solution doesn't make a lot of sense phyisically (it
basically means putting to charges with opposing charges on top of each
other), but why is it wrong mathematically? Or why do I lose solutions
when I do it that way?
I hope someone can help
Cheers