Solving polynomial equations by iteration - Quadratic Equations

Notes:

In this lesson, we will learn:

• Solving Equations Using Direct Iteration
• Evaluating equations Using Iteration by Bisection
• Analyzing Equations Using Newton-Raphson Method

• Iteration means to repeatedly solving an equation to obtain a result using the result from the previous calculation.
• Direct iteration:

1. Rearrange the original equation such that the term in which the variable with the highest exponent is isolated.

2. Leave the variable on its own on the LHS by performing inverse operation.

3. The LHS becomes $x_{n+1}$.

4. The RHS becomes $x_n$.
• Iteration by bisection:

1. Shrink the interval where the roots lies within 2 equal parts.

2. Decide in which part the solution resides.

3. Repeat the steps until a consistent answer is achieved.
• Newton-Raphson method:
$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$