what is the engineering applications of roots of equations by numerical methods?

Answer 1

Here is one application in electrical engineering:

The transfer function of a circuit can be represented in the complex s domain as the ratio of two polynomials in s. The roots of the numerator polynomial are the "zeros" of the function and the roots in the denominator are the "poles" of the function. The location of the pole and zeros determine the transient and steady-state response of the circuit.

Answer 2

The first thing that comes to mind is poles and nodes of filters.