Calculating the point where potential V = 0 (due to 2 charges)

[link223]

Homework Statement

Two point charges, and are placed
5.0 cm apart on the x axis. At what points along the x axis is
(a) the electric field zero and (b) the potential zero? Let
V = 0 at r = infinity.

Relevant Equations

Electric potential

Apparently, there are two solutions where the electric potential is zero which I don't understand, can I get some input on how this is possible?
I have one thing in mind (which I just thought of and might solve it), the equipotentiality i.e. when I draw a circle for V = 0 around the negative charge it will have 2 points where V = 0 one to the right and one to the left because the equipotential lines are continuous. Is that reasoining correct?

 

[berkeman]

I'm not able to read your photo very well. The problem statement does not say what the magnitudes of the 2 charges are.

Also, please have a look at the "LaTeX Guide" link below the Edit window. Posting math equations in LaTeX is *much* more readable that posting dim photos of your hand-written work (and is required by the PF rules). Thanks.

 

[haruspex]

Homework Statement:: Two point charges, and are placed
5.0 cm apart on the x axis. At what points along the x-axis is
(a) the electric field zero and (b) the potential zero? Let
V = 0 at r = infinity.
Relevant Equations:: Electric potential

Apparently, there are two solutions where the electric potential is zero which I don't understand, can I get some input on how this is possible?
I have one thing in mind (which I just thought of and might solve it), the equipotentiality i.e. when I draw a circle for V = 0 around the negative charge it will have 2 points where V = 0 one to the right and one to the left because the equipotential lines are continuous. Is that reasoining correct?
View attachment 301419

There must be two solutions in principle because you got a quadratic, and your argument is basically sound, but there will be degenerate cases. E.g. if the charges are equal and opposite then one solution goes to infinity.