Case bb(a=c=0)
If a=c=0 then any value of x will satisfy the equation, so there will be an infinite number of solutions.
color(white)()
Case bb(a=0, c!= 0)
If a=0 and c!=0 then the left hand side of the equation will always be 0 and the right hand side non-zero. So there is no value of x which will satisfy the equation.
color(white)()
Case bb(a!=0, c=0)
If a != 0 and c=0 then there is one solution, namely x=0.
color(white)()
Case bb(a > 0, c > 0) or bb(a < 0, c < 0)
If a and c are both non-zero and have the same sign, then there are two Real values of x which satisfy the equation, namely x = +-sqrt(c/a)
color(white)()
Case bb(a > 0, c < 0) or bb(a < 0, c > 0)
If a and c are both non-zero but of opposite sign, then there are no Real values of x which satisfy the equation. If you allow Complex solutions, then there are two solutions, namely x = +-i sqrt(-c/a)
