/sci/ - Science & Math

Why can we divide by [math] X'(x)+X(x) [/math] in the third to last line?
I understand that we can divide by [math] X(x) [/math] and [math] T(t) [/math] as we assume they are non-zero to avoid a trivial solution, but i'm not sure why the above is possible.
in fact if [math] X(x)=\alpha e^{-x} [/math], which is always non-zero, then [math] X'(x)+X(x)=0 [/math].