# Root Finding

Root finding refers to the general problem of searching for a solution of an equation $F(x)=0$ for some function $F(x)$. This is a very general problem and it comes up a lot in mathematics! For example, if we want to optimize a function $f(x)$ then we need to find critical points and therefore solve the equation $f'(x)=0$.

There are few examples where there exist exact methods for finding solutions. For example, the quadratic formula

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

gives us an exact method for finding roots of the equation

$$ ax^2 + bx + c = 0 $$

There is a general formula to solve a cubic equation and even a quartic (degree 4) equation (but the formula is too complicated to be useful).

But there does not exist a formula for a quintic (degree 5) polynomial. And there are *many* more examples of equations with no known method to solve them exactly.

What can we do? Use numerical methods to find approximate solutions.