Root Finding Methods
On this section, I will briefly describe finding the root of an equation using the Bisection Method and Newton-Raphson Method.
What is the Root of an Equation?
The root of an equation is a value that satisfies the equation, meaning that when the root is substituted into the equation, it results in a true statement (usually equating to zero). In other words, the root is the value of the variable that makes the equation equal to zero.
Example
Consider the quadratic equation:
[ f(x) = x^2 - 4x + 3 ]
To find the roots of this equation, we solve for ( x ) such that ( f(x) = 0 ).
The equation can be factored as:
[ (x - 1)(x - 3) = 0 ]
From this factorization, we can see that the roots of the equation are:
[ x = 1 ] and [ x = 3 ]
These are the values that satisfy the equation ( x^2 - 4x + 3 = 0 ).
Next, we will delve into two numerical methods to find roots of equations: the Bisection Method and the Newton-Raphson Method.