how do you know if a quadratic equation will have 1,2 or no solutions

Well, it always has two solutions.

Of course they may be identical, in which case the vertex is on the axis of the squared variable.
Or they may be complex which means the whole parabola never crosses that axis.
If form is
y = a x^2 + b x + c
then look at
b^2-4ac
If that is positive, two real roots
If that is zero, two identical roots with vertex on y = 0
If that is negative, then no real roots but two complex roots.