H2 Maths equations and inequalities formula sheet: discriminant cases for quadratics, completing the square, modulus inequality rules, rational inequality sign-analysis, and the...
Q: What does H2 Maths Notes (JC 1-2): 1.3) Equations and Inequalities cover? A: Algebraic, graphical, and numerical techniques for solving equations and inequalities in H2 Maths Topic 1.3.
Before you revise Practise each solving technique with and without technology. Mark schemes expect both algebraic reasoning and clear statements of graphing calculator (GC) commands used when numerical methods are required.
Solving means finding allowed values: Check the domain first.
Equations find roots; inequalities need intervals: Use factorisation, graphs, or sign charts.
Numerical answers still need method and restrictions: State the GC command, interval, or iteration rule used.
. Move everything to one side, build a sign chart, and exclude
x=−3
.
Status: SEAB's current H2 Mathematics (9758) syllabus PDF is labelled for 2026. Topic 1.3 expectations are within Section A Pure Mathematics, which is assessed in Paper 1 (100 marks) and Paper 2 Section A (40 marks).
Formulas at a glance
Every key result and method the 9758 syllabus expects you to apply, collected on one screen. Worked examples for each appear in the sections below.
Quadratic equations
Result
Rule / method
Discriminant
b2−4ac>0: two distinct real roots; b2−4ac=0: one repeated root; b2−4ac<0: no real roots
Quadratic formula
x=2a−b±b2−4ac
Completing the square
ax2+bx+c=a!(x+2ab)2−4ab2−4ac
Modulus inequalities
Type
Rule
∣x∣<a (a>0)
−a<x<a
∣x∣>a (a>0)
x<−a or x>a
∣f(x)∣<a
−a<f(x)<a; solve the resulting compound inequality
∣f(x)∣>a
f(x)<−a or f(x)>a
Rational inequalities
Type
Rule / method
Sign-analysis method
Move all terms to one side, form Q(x)P(x)≷0, find critical points from numerator and denominator, then sign-test each interval
Multiply-by-square trick
Multiply both sides by [Q(x)]2>0 to clear the denominator without flipping the inequality sign; equivalent to forming P(x)⋅Q(x)≷0 when Q(x)=0
Endpoint rule
Exclude values that make the denominator zero; include values where the expression equals zero only for non-strict inequalities (≤ or ≥)
Solving methods
Method
When to use
Factorisation / formula
Polynomial equations with rational or surd roots
Completing the square
Turning-point form, or when the formula is not needed
Substitution
Equations reducible to a quadratic (e.g. u=x2)
Graphical / GC
f(x)=g(x) with no clean algebraic solution; plot both curves and find intersections
Fixed-point iteration
Rearrange to x=g(x); converges when ∣g′(x)∣<1 near the root
Newton-Raphson
xn+1=xn−f′(xn)f(xn)
Equation-Solving Toolkit
Exact algebraic methods: factorisation, substitution, completing the square, and standard formulae.
Graphical methods: plot y=f(x) and y=g(x); solutions satisfy f(x)=g(x).
Iterative methods: rearrange into x=g(x) or use Newton-Raphson xn+1=xn−f′(xn)f(xn)
Numerical equation solver: graphing calculator (GC) features (SOLVE, ISCT, nSolve) find roots when analytic solutions are messy; document the initial guess or interval.
Example -- Iteration
Solve x=cosx.
Define g(x)=cosx; iterate xn+1=g(xn).
With x0=0.7, the sequence converges to 0.739085133….
State the termination criterion (difference less than 10−4).
Convergence note: here g′(x)=−sinx so ∣g′(x)∣=∣sinx∣<1 near the fixed point, which ensures local convergence of the iteration.
Polynomial and Rational Equations
For quadratics, use the discriminant b2−4ac to comment on nature of roots.
Cubics: attempt factor theorem or GC ISCT to guess rational roots before long division.
Rational equations require clearing denominators carefully and checking for extraneous roots.
Example -- Rational equation
Solve x−1x+2=3.
Multiply both sides by x−1: x+2=3x−3.
Simplify to 2x=5⇒x=25.
Ensure x=1 (domain restriction).
Root-check checkpoint
Use this check after any algebraic move that changes the original equation or inequality.
Algebraic move
What can go wrong
Check before final answer
Clear a denominator
A candidate value may make the original denominator zero.
List restrictions before solving, then reject any root outside the domain.
Square both sides
The squared equation also accepts the case where the two original sides have opposite signs.
Substitute each candidate root into the original equation.
Cancel a factor
A root from the cancelled factor may be lost.
Factor first and set each factor to zero before dividing.
Multiply an inequality by an unknown expression
The inequality sign may need to flip when the expression is negative.
Move everything to one side and use a sign chart instead.
Worked check: if squaring gives candidate roots x=1 and x=4, do not report both immediately. Substitute each into the original equation. If x=1 makes a denominator zero or makes the unsquared sides have opposite signs, reject it and state why.
Inequalities
Linear and Polynomial Forms
Rearrange to f(x)>0 or f(x)≤0 and sign-test intervals around roots.
Quadratics: identify roots α,β and sketch or use sign chart.
Example -- Quadratic inequality
Solve x2−5x+6≥0.
Factor: (x−2)(x−3)≥0.
Sign chart shows positive on (−∞,2]∪[3,∞).
Rational Inequalities
Identify critical points from numerator and denominator.
Construct sign tables including excluded values.
Express solution sets using interval notation.
Example -- Rational inequality
Solve x+32x−5<1.
Rearrange: x+32x−5−1<0 (simplify to) x+3x−8<0.
Critical points at x=−3 (excluded) and x=8.
Sign chart yields solution (−3,8). Both endpoints are excluded: x=−3 makes the expression undefined, while x=8 makes the expression equal to 0 rather than less than 0.
Endpoint classification checkpoint
Before writing a final inequality interval, classify every critical value by what it does in the original expression. This prevents a correct sign chart from turning into wrong bracket notation.
Critical value type
What happens there
Bracket decision
Common trap
Denominator is zero
Original expression is undefined.
Always exclude.
Closing the bracket because the sign chart touches the boundary.
Numerator is zero and inequality is strict
Expression equals zero.
Exclude for < or >.
Including the root just because it is a solution of the related equation.
Numerator is zero and inequality is non-strict
Expression equals zero.
Include for ≤ or ≥, unless another restriction excludes it.
Forgetting that equality is allowed.
Squared factor touches the axis
Sign may not change across the point.
Decide inclusion from the inequality sign and domain, not from sign change alone.
Assuming every critical point flips the sign.
Worked check: for x+1(x−2)2≥0, the critical values are x=−1 and x=2. Exclude x=−1 because the denominator is zero. Include x=2 because the expression equals 0 and the inequality allows equality. The sign does not change at x=2, but the point still belongs in the solution set.
Misconception check: open or closed brackets are not chosen by how the graph looks near the endpoint. They come from the original expression, the inequality sign, and any domain restriction.
Inequalities with Modulus and Exponentials
Split modulus cases: ∣f(x)∣<a implies −a<f(x)<a.
For exponentials/logarithms, apply monotonicity: ax>ay⟺x>y for a>1.
Simultaneous Equations
Linear systems: solve via elimination, substitution, or matrix methods. Express solutions clearly (point, line, or no solution).
Linear-quadratic or quadratic-quadratic systems: substitute solutions from one equation into the other.
Example -- Line-circle intersection
Find intersections of y=2x−1 with x2+y2=13.
Substitute: x2+(2x−1)2=13 giving 5x2−4x−12=0.
Solve: x=104±16+240=104±256=104±16
Hence the intersection points are (2,3) and (−56,−517)
Numerical Methods and Technology
Newton-Raphson: ensure derivative f′(xn)=0; document iteration formula and starting value.
Fixed-point iteration: check convergence using ∣g′(x)∣<1 near the root.
State graphing calculator (GC) commands (e.g. Solver, nSolve, G-Solv) and the viewing window or guesses used.
Example -- Newton-Raphson
Find a root of f(x)=x3−x−1 to 3 significant figures.
Want weekly guided practice on Equations and Inequalities? Our H2 Maths tuition programme builds fluency in this topic through structured problem sets and exam-style drills.
Common exam mistakes
Multiplying both sides of an inequality by an unknown expression: If you multiply g(x)f(x)>0 by g(x) without considering its sign, you risk flipping the inequality in cases where g(x)<0. Always multiply by [g(x)]2>0 instead, or split into sign cases.
Squaring both sides without checking validity: Squaring f(x)=g(x) introduces the equation [f(x)]2=[g(x)]2
Cancelling a common factor that could be zero: Dividing both sides by (x−a) loses the solution x=a. Factorise and set each factor to zero rather than cancelling.
Forgetting to exclude restricted values in rational expressions: After solving a rational equation, verify that no solution makes any denominator zero - these values are outside the domain and must be rejected.
Incorrect interval notation for inequality solution sets: Writing x∈[2,3] when the endpoints must be excluded (e.g. from a strict inequality or a domain restriction) is a common mark-loss. Use open brackets (2,3) where appropriate and state excluded values explicitly.
Frequently asked questions
Which papers assess Equations and Inequalities in the 9758 A-level exam? Topic 1.3 falls under Pure Mathematics, which is examined in both Paper 1 (100 marks, pure only) and Paper 2 Section A (40 marks, pure). [1] Questions on solving equations, inequalities, and numerical methods can appear in either paper, so practise under both time constraints.
Can I use the graphing calculator (GC) to solve inequalities, or must I show algebraic working? You may use the GC to verify or locate solutions, but the mark scheme expects full algebraic working - a sign chart or rearranged inequality showing critical points and tested intervals. Quoting only a GC answer without supporting reasoning will not earn method marks. State the GC command and window settings used when numerical methods are required.
Is solving a system of linear equations with three unknowns part of the 9758 syllabus? Yes. SEAB 9758 explicitly includes systems of linear equations in three unknowns. [1] You are expected to solve them using the GC (via matrix row reduction or a simultaneous-equation solver) and to interpret the solution geometrically - whether the three planes meet at a unique point, along a line, or have no common intersection.
Is there a formula sheet for H2 Maths equations and inequalities? Yes - the "Formulas at a glance" section near the top of this page collects every key result and method you need: discriminant cases, completing the square, the modulus inequality rules (∣x∣<a⟺−a<x<a and ∣x∣>a⟺x<−a or x>a), the sign-analysis and multiply-by-square-of-denominator approaches for rational inequalities, and the full menu of solving methods including the graphical/GC approach. A downloadable PDF version is also linked at the top of this page.
Other H2 Maths formula sheets
Revising more than one topic? Grab the matching one-page formula sheet:
SEAB H2 Mathematics syllabus (9758), examinations from 2026 - Topic 1.3 Equations and inequalities, including solution of equations/inequalities and numerical methods, under Section A Pure Mathematics. See the citation entry above for the official PDF URL.
