Locating Roots of an Equation — Numerical Methods (9709 P3) — Mathematics

Introduction

Many equations that arise in A-Level Mathematics — such as $x^3 - 3x + 1 = 0$ or $e^x = 4 - x$ — cannot be solved by exact algebraic methods. Numerical methods provide systematic techniques for finding approximate solutions to any desired degree of accuracy.

Before refining an approximation, you must first know where to look. This subtopic covers the two key strategies for locating a root approximately:

Graphical considerations — sketching or interpreting graphs to identify where curves intersect.
Sign-change search — evaluating a function at successive points and detecting where it changes sign.

These techniques appear in almost every 9709 P3 numerical methods question, typically as the opening part worth 1–2 marks.

Core Concept

What is a root?

A root of an equation $f(x) = 0$ is a value $x = \alpha$ such that $f(\alpha) = 0$ . Geometrically, a root is an $x$ -coordinate where the curve $y = f(x)$ crosses or touches the $x$ -axis.

Graphical method

Rearrange the equation into the form $g(x) = h(x)$ (often one curve is simpler), then sketch both curves on the same axes. Each intersection point corresponds to a root. The sketch identifies the approximate integer region in which each root lies.

Sign-change method

If $f(x)$ is continuous on an interval $[a, b]$ , and $f(a)$ and $f(b)$ have opposite signs, then by the Intermediate Value Theorem there is at least one root in $(a, b)$ .

f(a) \cdot f(b) < 0 \implies \exists\, \alpha \in (a,b) \text{ such that } f(\alpha) = 0

To locate a root between consecutive integers $n$ and $n+1$ , evaluate $f(n)$ and $f(n+1)$ ; if one is positive and the other negative, a root lies in that interval.

Key condition: The sign-change test is only valid for continuous functions. If $f$ has a vertical asymptote or a discontinuity in $[a,b]$ , a sign change does NOT guarantee a root — it may indicate a discontinuity instead.

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