A Level · Cambridge (CIE) · 9709

Locate equation roots with graphs and valid sign changes

Turn an equation into f(x) = 0, inspect the graph and bracket a root only where the original function is continuous.

Section 2.6 · Numerical solution of equations

Pure Mathematics 2 & 3 · Numerical solution of equations · 2.6 / 3.6

Exam scope and assessment notes

Pure Mathematics 2 (Paper 2, AS only) and Pure Mathematics 3 (Paper 3, A Level) · Mathematics 9709, 2026–2027. Paper 1 graph and algebra knowledge is assumed. The AS route using Papers 1 + 2 cannot be carried forward to A Level. Shared sections 2.6 and 3.6 include locating roots graphically or by sign changes, notation for convergent approximation sequences, and using a given iteration or rearrangement to a prescribed accuracy. An iteration may fail to converge. Knowledge of the general condition for convergence is not required here. Check the original equation, domains and requested rounding. Written practice is self-checked.

Before you begin

Evaluate a function accurately and read graph intersections.
Identify polynomial, logarithmic and reciprocal domains.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

What do opposite non-zero signs at the endpoints imply for a continuous function?

What you will learn

Locate approximate roots from axis crossings or intersections of two graphs.
Use opposite signs to find a root interval while recognising touching roots and false brackets across poles.

The blue graph is f(x) = x³ − x − 1 in the window 1 ≤ x ≤ 1.6. At 1.3 its value is −0.103 and at 1.4 it is 0.344. The pale vertical band marks this continuous sign-change bracket, so a root lies between 1.3 and 1.4. The graph locates the positive crossing but its drawing precision does not justify several decimal places. The vertical plotting scale is shown at the left edge; the actual y axis x = 0 lies outside this window. The curve continues beyond the window.

Find where the original equation becomes zero

Write the equation as f(x) = 0 without changing its valid inputs. Roots appear where the graph of f meets the x axis. For an equation g(x) = h(x), intersections of their two graphs give the same root inputs as zeros of f(x) = g(x) − h(x). A sketch can locate a plausible interval, but its drawing resolution does not usually justify several decimal places.

For f(x) = x³ − x − 1, f(1) = −1 and f(2) = 5. The polynomial is continuous throughout [1, 2], so it has at least one root strictly between those consecutive integers. Values f(1.3) = −0.103 and f(1.4) = 0.344 refine the positive root's location to (1.3, 1.4). A sign change establishes existence on a continuous interval; uniqueness needs separate information about the function, such as its increasing behaviour there.

A sign change needs continuity, and its absence does not exclude every root

For f(x) = 1/(x − 1), f(0) is negative and f(2) positive, but there is no root: the sign changes across an undefined pole. This interval fails the continuity requirement. Retain all original denominator, square-root and logarithm restrictions before using a sign-change argument. If an endpoint itself gives f(a) = 0, that endpoint is already an exact root rather than a strict interior sign-change bracket.

For f(x) = (x − 1)², the graph touches the axis at x = 1 and remains non-negative. Values at 0 and 2 have the same positive sign despite the root between them. Thus no sign change does not prove no root. Use the graph and the equation together, especially for touching or repeated roots. Numerical sampling over one region also does not by itself establish that every root of the equation has been found.

Pause and explain

Does 1/(x − 1) have a root between 0 and 2 because its endpoint signs differ?

Put the idea to work

Worked example

Locate a root of x³ − x − 1 = 0 between consecutive integers, then refine the interval using 1.3 and 1.4.

Show the worked solution

Evaluate f(1) = −1 and f(2) = 5.
The polynomial is continuous, so the opposite signs locate at least one root in (1, 2).
Evaluate f(1.3) = −0.103 and f(1.4) = 0.344, giving the narrower bracket (1.3, 1.4).

Answer A root lies in (1, 2), more precisely in (1.3, 1.4).

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

Locate both roots of x² − 2 = 0 between consecutive integers using signs.

Give me a hint

Check one negative-input bracket as well as one positive-input bracket.

Compare my reasoning

At −2 and −1 the values are 2 and −1, so a root lies in (−2, −1).
At 1 and 2 the values are −1 and 2, so a root lies in (1, 2).
The polynomial is continuous on both intervals; the two distinct brackets locate the two roots of this quadratic.

Roots in (−2, −1) and (1, 2).

Look for these in your work

Evaluate signs on both relevant input regions.
Use continuity to support each root bracket.

2 · Independent

For f(x) = (x − 1)², identify the root and explain why testing the signs at 0 and 2 misses it.

Give me a hint

Factor or inspect the original square and describe how its graph meets the axis.

Compare my reasoning

The function is zero precisely when x − 1 = 0, so x = 1 is a root.
Both endpoint values are 1, with the same positive sign.
The upward parabola touches the axis at its repeated root; absence of a sign change does not exclude this contact.

Root x = 1; same-sign endpoint values miss the touching root.

Look for these in your work

Find the root from the original expression.
Explain the limitation of sign-change-only searching.

3 · Transfer

A supplied index model is R(t) = eᵗ + t, with t measured numerically in hours and t ≥ 0. Locate a time when R reaches 3 between consecutive whole hours.

Give me a hint

Use f(t) = eᵗ + t − 3 and check its continuous positive-time interval.

Compare my reasoning

At t = 0, f(0) = −2.
At t = 1, f(1) = e − 2 > 0.
The function is continuous, so a threshold time lies strictly between 0 and 1 hours; this bracket does not yet specify decimal places.

A threshold time lies in 0 < t < 1 hours.

Look for these in your work

Translate the threshold into a valid zero equation.
Use a continuous sign-change bracket without claiming unsupported precision.

Common mistakes

Check continuity throughout the proposed bracket, not only endpoint values.
A graph locates roots; its visual precision does not certify decimal places.
Same-sign endpoint values can still surround a touching root.

Recall without your notes

What can you explain now?

For (x − 1)², does a positive value at both x = 0 and x = 2 prove there is no root between them?

Compare with the explanation

No; x = 1 is a touching root.

The graph meets the axis without crossing it, so there is no sign change across that repeated root.

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Make it stick

Explain how zeros and graph intersections locate equation solutions. Give one sign change across a pole that fails and one touching root without a sign change.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

Keep learning Mathematics

Check the official syllabus 2026–2027

Course scope and assessment

312 sequenced lessons provide 936 guided, independent and application problems covering the 160 objectives in sections 1.1–1.8, 2.1–2.6, 3.1–3.9, 4.1–4.5, 5.1–5.5 and 6.1–6.5. The map lists all 38 sections across six alternative components; all eight Pure Mathematics 1 sections and all six Pure Mathematics 2 sections, from Paper 2 Algebra through Numerical solution of equations, currently have full teaching sequences. Paper 3 Algebra, section 3.1, links the six shared foundation lessons plus six lessons on partial fractions and rational binomial expansions; Paper 3 logarithms, trigonometry and numerical equations use the shared lessons, with the Paper 1 graph foundation linked into trigonometry. Paper 3 differentiation combines the shared methods with its inverse-tangent derivative lesson. Paper 3 Integration links four shared standard and identity lessons plus seven advanced lessons, covering all six section 3.5 requirements. Eight Vectors lessons cover all six section 3.7 requirements, with 2D and 3D line and solid applications. Nine Differential equations lessons cover all four section 3.8 requirements, including Paper 3 integration techniques, initial conditions, equilibrium solutions and conditional model interpretations. Eleven Complex numbers lessons cover all eight section 3.9 requirements, with seven Argand diagrams and explicit argument and boundary checks. All nine Pure Mathematics 3 sections now have teaching linked to every mapped requirement. Nine Forces and equilibrium lessons cover all seven section 4.1 requirements, with six force diagrams, signed incline-friction intervals, reaction-dependent pull and push limits and separate-body third-law checks. Nine Kinematics lessons cover the four official section 4.2 requirements, with the graph requirement mapped to three relationships. Six motion diagrams support directed quantities, graph slopes and signed areas, Paper 1 time-calculus methods, reversal distances, constant-acceleration stages and shared-clock particle meetings. Six Momentum lessons cover both section 4.3 requirements, including signed system totals, conservation assumptions, outgoing velocities, coalescence, unknown masses and successive impacts. Four diagrams show system momentum through the stipulated events, and supplied kinetic-energy checks distinguish conservation of momentum from changes in kinetic energy. Twelve Newton’s laws lessons cover all four section 4.4 requirements, with vertical support and release stages, rough horizontal and incline motion, inferred parameters, hanging and surface pulley systems, rigid tow-bar thrust and slack-string transitions. Seven force diagrams support the separate-particle balances. Nine Energy, work and power lessons cover all five section 4.5 requirements, with angle-dependent work, kinetic and reference potential energy, signed net-work balances, smooth curved slides, rough and driven slopes, ideal connected-system energy and instantaneous power on hills. Five diagrams distinguish the energy inputs, losses and valid positive-speed force model. All five Mechanics sections now have teaching and staged practice linked to their mapped requirements. Thirteen Representation of data lessons cover all five section 5.1 requirements, including choice and limitations of displays, ordered and back-to-back stems, raw summaries and box plots, unequal-width density histograms, cumulative-frequency interpolation, descriptive moment calculations, grouped estimates, coding, combining two datasets and correcting totals. Six accessible diagrams retain repeated observations and distinguish numerical spread, frequency area and estimated readings. Ten Permutations and combinations lessons cover both official section 5.2 requirements, with selection and ordered roles, repeated objects, adjacency blocks, non-adjacency gaps, leading-zero cases, labelled multi-row seating and mixed restrictions. Six accessible diagrams show outcome trees, ordered versus unordered choices, repeated copies, blocks, gaps and distinct row seats. Eleven Probability lessons cover all four section 5.3 requirements, with equiprobable elementary outcomes, selection-based probabilities, complements and disjoint regions, independence tests, restricted sample spaces, two-way tables, replacement and removal trees, reversed conditions and unknown parameters. Seven accessible diagrams show dice outcomes, random subsets, event regions, a conditioned grid, a count table and two probability trees. Twelve Discrete random variables lessons cover all three official section 5.4 requirements, with complete probability tables, moment and parameter checks, binomial assumptions, exact probabilities and moments, geometric first-success probabilities, tails, expectation and integer reliability thresholds. Six accessible diagrams distinguish outcome grouping, weighted moments, fixed trials, discrete probability bars, waiting conventions and integer tail deadlines. Twelve Normal distribution lessons cover all three official section 5.5 requirements, with continuous-area interpretation, MF19 cumulative tables, full standardisation, tail and interval regions, conditional probabilities, percentiles, one- and two-parameter inference, binomial suitability, continuity corrections and approximate integer thresholds. Seven accessible diagrams show model areas, table reading, aligned scales, intervals, quantiles, matched binomial and normal shapes and whole integer cells. All five Statistics 1 sections now have teaching and staged practice linked to their mapped requirements. Fourteen Poisson distribution lessons cover all five official section 6.1 requirements, with homogeneous event-count assumptions, exposure scaling, exact masses and tails, conditional count events, moments, parameter inference, probability duration targets, disjoint and overlapping windows, rare-count binomial approximations, matched normal approximations and continuity corrections. Nine accessible diagrams show event exposures, unit conversions, discrete bars, complete tail complements, population moments, shared windows, approximate count models and integer cells. Exact checks distinguish approximate and exact close-target decisions. Fourteen Linear combinations lessons cover the six listed results in official section 6.2, with affine moments, signed scales and constant cases, expectation without independence, independent squared variance contributions, fixed and random fees, independent copies versus reused values, affine normal events, independent normal sums and differences, weighted and conditional areas, coefficient inference, exact independent Poisson sums and overlapping exposure limits. Nine accessible diagrams distinguish scale changes, variance contributions, component costs, copy mechanisms, aligned normal scales, summed areas, Poisson count pairs and unsupported distribution claims. Fourteen Continuous random variables lessons cover both official section 6.3 requirements, with density validity and units, support-aware intervals, complements and conditional regions, first and second moments, percentiles by direct density areas, symmetry checks, parameter inference, infinite-support power and exponential integrals, divergent-moment limits and combined applications. Nine accessible diagrams distinguish density height from probability, normalisation, negative-density failures, central and conditional areas, mean versus median, percentile locations and partial infinite tails. Explicit cumulative distribution function knowledge is not required. Eighteen Sampling and estimation lessons cover all eight official section 6.4 requirements, with population and frame definitions, random-number selection, sampling-quality limits, independent sample-mean moments, normal means, CLT conditions, SE scaling, raw, summarised and coded unbiased estimates, known-variance and large-sample mean intervals, repeated-sampling interpretation, precision planning, proportion intervals, sparse-outcome limits and contextual applications. Ten accessible diagrams distinguish selection frames, number rejection, repeated sample means, population versus mean density, CLT models, variance divisors, interval targets, illustrative coverage, integer size checks and proportion margins. Eighteen Hypothesis tests lessons cover all five official section 6.5 requirements, with pre-specified alternatives, inclusive count tails, explicit equal-tail conventions, integer critical regions and actual significance, exposure-aware Poisson tests, continuity-corrected count approximations, normal and large-sample mean tests, confidence-interval connections, error classifications, fixed-region binomial, Poisson and normal error probabilities and sample-size power comparisons. Nine accessible diagrams show the testing procedure, exact count regions, unequal Poisson tails, discrete significance jumps, corrected count boundaries, matching mean intervals and alternative error distributions. All shared lessons are counted once in the course total. All 38 sections now link teaching and practice. Eight earlier overview lessons remain available, but do not count as full section coverage. This is a content-availability measure, not full programme completion or exam readiness. Written practice is self-checked, not officially marked. Check the 2028–2030 syllabus separately if sitting an examination from 2028; this teaching map uses 2026–2027 requirements.