Discriminant

Question 1

b^2- 4ac

Answer 1

One real solution, curve meets the tangent at one point

Question 2

b^2- 4ac =0

Answer 2

There are no real solutions, and the tangent doesn’t meet the curve at any point

Question 3

b^2- 4ac >0

Answer 3

There are two distinct real solutions and the tangent crosses the curve at two points