"Root" is a more formal way to say "answer" or "solution". A number is called a root of an equation if when the number is substituted into the equation and both sides simplified, the result is an identity, such as 2=2 or 8=8, etc.
Or
The root of a one dimensional equation is the value of x for which the equation y=f(x)=0.
Example 1
There are two roots to this equation x²+6=5x
They are 3 and 2.
3 is one root because when you substitute 3 for x
3²+6=5(3)
9+6=15
15=15
Also 2 is a root, because when you substitute 2 for x
2²+6=5(2)
4+6=10
10=10
Example 2
The root value for the given equation y = f(x) = 10x2 + 31x - 14 is "0.4"
because if we substitute the root value "0.4" to the equation and we get zero.
(10 * (0.4)^2) + (31 * 0.4) -14 = 0
WHY
One might ask what this has to do with chemistry or science, etc. There are many examples of chemical problems in which one needs to find the roots of equations. Some of the chemical examples include
- · Weak acid/base equilibria; titrations
- · The vapor pressure equation, ln P = A + (B/T) + C ln T
- · Equation of state problems
- · Equilibrium calculations
- · Quantum mechanics
- · Kinetics
HOW
So, now is the time to ask how do we fine the roots for the given equation?
1. Newton’s method (Newton-Raphson method)
Newton’s method consists of four steps:
1. Guess a value of the root, x.
2. Calculate f(x) and the derivative f’(x), i.e., df(x)/dx
3.
4. Iterate
[to be continued...]
