Numerical Solution of Equations

2026 What You Need to Know (Syllabus Objectives)

By the end of this topic, you should be able to:

  1. Locate approximately a root of an equation by looking at graphs and/or searching for a sign change (for example, finding two consecutive whole numbers between which a root lies)

  2. Understand the idea of a sequence of approximations that gets closer and closer to a root, and use the correct notation for this

  3. Understand how iterative formulas work (formulas in the form x_{n+1} = F(x_n)), how they relate to the equation being solved, and use them to find a root to a given level of accuracy. You also need to understand that iterations can sometimes fail to converge (fail to get closer to an answer)

Introduction to Numerical Methods

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Sometimes we need to solve equations that cannot be solved using normal algebraic methods (like factoring or using a formula). For example, equations like x³ - x² - 9 = 0 or xe^x - 2x³ - 0.5 = 0 are very difficult or impossible to solve exactly.

Numerical methods are techniques that help us find approximate solutions (close-enough answers) to these difficult equations. The two main methods you'll learn are:

  • Change of sign method – finding where a root is located
  • Iteration – repeatedly using a formula to get closer to the exact answer

A root of an equation is a value of x that makes the equation equal to zero. On a graph, roots are where the curve crosses the x-axis.

1. Change of Sign Method

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What is a sign change?

When you have a continuous function f(x) (a function with no breaks or jumps), and you find that:

  • f(a) is positive
  • f(b) is negative (or the other way around)

Then there must be a root between a and b. This is because the function has to cross the x-axis (where f(x) = 0) to change from positive to negative.

Example: If f(2) = 5 (positive) and f(3) = -2 (negative), then there must be a root between x = 2 and x = 3.

How to use the change of sign method

Step 1: Choose two values of x (let's call them a and b) that you think might have a root between them.

Step 2: Calculate f(a) and f(b) by substituting these values into your function.

Step 3: Check if the signs are different:

  • If one is positive and one is negative → there's a root between a and b
  • If both are positive or both are negative → no conclusion (there might be no root, or there might be multiple roots)

Example: Show that f(x) = x³ - 5x + 1 has a root between x = 2 and x = 3.

Solution:

  • f(2) = (2)³ - 5(2) + 1 = 8 - 10 + 1 = -1 (negative)
  • f(3) = (3)³ - 5(3) + 1 = 27 - 15 + 1 = 13 (positive)
  • There is a change of sign from negative to positive
  • Therefore, there must be a root between x = 2 and x = 3

Showing a root to a given accuracy

To show that a root is correct to a certain number of decimal places, you need to find its upper and lower bounds and show there's a sign change between them.

Example: Show that the root is 2.4 correct to 1 decimal place.

Solution:

  • A value of 2.4 to 1 decimal place could actually be anywhere from 2.35 up to (but not including) 2.45
  • So we check: f(2.35) and f(2.45)
  • f(2.35) = -0.12... (negative)
  • f(2.45) = 0.18... (positive)
  • There's a sign change, so the root is between 2.35 and 2.45
  • All values in this interval round to 2.4, so the root = 2.4 to 1 decimal place ✓

Important notes about sign change

This method only works when:

  • The function is continuous (no breaks, jumps, or asymptotes in the interval)
  • The interval is small (not too wide)

Always remember: You must show your working. Simply saying "there's a sign change" without calculating f(a) and f(b) will not get you marks.

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