Question

explain the difference between quadratic equations with one solution, two solutions, and complex solutions

Answer 1

A quadratic equation of the form ax^2 + bx + c has a general solution of the form x = - b +- b^2 - 4ac/2a
We shall use the expression under the radical in the formular method. The expression under the radical is called the discriminant.
If the discriminant D = b^2 - 4ac > 0 then the quadratic has two distinct real number solutions since the square root of any positive number is it self a positive number.
If D = b^2 - 4ac = 0 then we get expressions of the form (-b + 0) and (-b - 0). Regardless we end up with -b. This means when D = 0 we have one solution.
If D < 0 the quadratic equation has a conjugate pair of complex roots of the form a + bi