One positive, one negative solution => ray starting point is inside the sphere. Both solutions positive => ray intersects the sphere in the direction of the ray twice. Both solutions negative => sphere is behind the ray. One positive solution => ray is tangent to sphere. One negative solution => sphere is behind the ray (tangent in opposite direction of the ray).

When the ray in tangent to sphere, the square root term drops, explaining why there is only one solution for t.

There can also be a solution where there are no intersections, in which I presume the math would show a negative under the square root, giving only complex solutions.

What is the physical interpretation of getting complex numbers for non intersections?

It possibly could be points on the complex unit sphere.