Why use methods as Newton, ridder or secant method for root finding?

Question

Why use methods as Newton, ridder or secant method for root finding?

I am bit confused for what reason someone would use these method to determine the root of a function, as it can easily be determined by solving $f(x) = 0$?

Answer 1

Not every equation can be solved without numerical methods. A few examples of some equations that "look easy" to solve, but can't be solved without numerical methods (at least, not in the way that you're thinking).

• $x^5 - x + 1 = 0$
• $\cos x = x$
• $\tan x = x$
• $e^x = 2x + 2$

Also, we need numerical methods to give us the decimal expansion to the "algebraic" answer, provided such an answer exists.