The concept you are looking for here is algebraic closure. If you want "ax^2 + bx + c" to have numeric roots and be factorable for all coefficients, you have to expand your definition of number from real to complex.

Probably the most blatant physical example of this is modeling mass-spring-damper systems. The differential equations will map the physical coefficients to such a polynomial, and the roots of that polynomial determine the system's behavior. Without complex numbers, you wind up only being able to describe real exponential responses and don't have the vocabulary to describe the underdamped (and oscillating) system responses with complex exponentials in the results.