Locating Roots

This lesson covers methods for locating roots of equations — the sign change method, decimal search, and systematic approaches — as required by the Edexcel A-Level Mathematics specification (9MA0). You also need to understand the limitations of these methods.

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The Sign Change Method

The Principle

If f(x) is a continuous function and f(a) and f(b) have opposite signs (one positive, one negative), then there must be at least one root between x = a and x = b.

This is a consequence of the Intermediate Value Theorem.

In mathematical terms: If f(a) × f(b) < 0 (one is positive, one is negative) and f is continuous on [a, b], then there exists at least one value c in (a, b) such that f(c) = 0.

Example

Show that x³ - 3x + 1 = 0 has a root between x = 1 and x = 2.

f(1) = 1 - 3 + 1 = -1 (negative) f(2) = 8 - 6 + 1 = 3 (positive)