10.1PureCore

Locating Roots

Many equations cannot be solved exactly. The first step in a numerical approach is to locate a root: show that a continuous function changes sign across an interval, so a root must lie inside it.

24 min Video by Zeeshan Zamurred Numerical Methods

Edexcel A level Maths: 10.1 Locating RootsWatch the full walkthrough before the notes below.

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What you'll be able to do

Use the sign-change rule
Show a root lies in an interval
Understand when sign change fails
Connect roots to f(x) = 0

The sign-change rule

If $f$ is continuous on $[a, b]$ and $f (a)$ and $f (b)$ have $opposite signs$ , then there is at least one root of $f (x) = 0$ in $(a, b)$ .

f (a) f (b) < 0 \Rightarrow root in (a, b)

Continuous f with a sign change.

f (1) = - 1 < 0

f (2) = 5 > 0

2Sign change and f continuous, so a root lies in

(1, 2)

AnswerRoot in (1, 2).

Tip — Always state that f is continuous — the rule needs it.

When it fails

A sign change guarantees a root, but no sign change does NOT guarantee none (there may be an even number of roots). Also, if $f$ is discontinuous (e.g. an asymptote) a sign change may not indicate a root.

Formula recap

f (a) f (b) < 0 \Rightarrow root in (a, b)

Sign-change rule (f continuous).

Common mistakes to avoid

Forgetting to check continuity.

The sign-change rule requires f continuous on the interval.

Concluding “no root” from no sign change.

There could be an even number of roots; no sign change is inconclusive.

Key takeaways

Sign change of continuous f over [a,b] ⇒ root in (a,b).
State continuity explicitly.
No sign change is not proof of no root.

Test yourself

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