In a real problem you can have equations with parameters, where coefficients depend on some other parameters. And you should be able to answer questions about the roots depending on parameter values, like 'in which range of parameters p you have real (non-complex) roots'? Having a formula for the roots based on coefficients lets you answer such questions easily. For example, for quadratic equation ax^2 +bx + c = 0 we have D = b^2 - 4ac, and if D < 0 then there is no real (non-complex) roots.