IGCSE Mathematics: Algebra and Graphs — Revision Notes

Condensed for the final weeks. For the full explanation, use the Algebra and Graphs study guide.

Letters and substitution

A letter stands for a number that can vary, a generalised number. If a pen costs p cents and a ruler costs r cents, 3 pens and 2 rulers cost 3p + 2r cents, whatever the prices are. If n is any integer, 2n is always even and 2n + 1 is always odd.

To substitute, replace each letter with its value, using brackets around negative numbers, then follow the order of operations:

v = u + at   with u = 12, a = -3, t = 5:   v = 12 + (-3)(5) = 12 - 15 = -3

Manipulation toolkit

Skill	Rule / reminder
Expanding	(a + b)(c + d) = ac + ad + bc + bd; (a + b)² = a² + 2ab + b² — not a² + b²
Factorising: common factor	Take out the highest common factor first, always
Factorising: difference of squares (Extended)	a² − b² = (a + b)(a − b)
Factorising: quadratic trinomial (Extended)	x² + (p+q)x + pq = (x + p)(x + q)
Indices	aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; a⁰ = 1; a⁻ⁿ = 1/aⁿ; (Extended) a^(1/n) = ⁿ√a
Algebraic fractions (Extended)	Find a common denominator before adding/subtracting, same as with numbers

Solving equations

• Linear: collect terms, then isolate the unknown — do the same operation to both sides.
• Simultaneous (linear): eliminate one variable by substitution or by adding/subtracting scaled equations.
• Quadratic (Extended) — three methods, in order of preference: solving a quadratic equation by any method is Extended-only content.
  1. Factorise first if it looks clean: x² + (p+q)x + pq = 0 → (x+p)(x+q) = 0 → x = −p or x = −q.
  2. Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a, for ax² + bx + c = 0.
  3. Completing the square: rewrite as a(x + h)² + k, useful for finding a turning point as well as solving.
• Changing the subject: treat the formula like an equation — whatever you do to isolate the new subject, do to both sides, in the same order you’d solve an equation for it.
• Simultaneous linear + quadratic (Extended): substitute the linear equation into the quadratic one, then solve the resulting quadratic.

Inequalities

• On a number line: open circle (○) for < or >, filled circle (●) for ≤ or ≥.
• Solving a linear inequality (Extended): solve exactly like an equation, except: multiplying or dividing both sides by a negative number reverses the inequality sign.
• Two-variable regions (Extended): draw the boundary line (solid for ≤/≥, broken for </>), then shade the unwanted side, the syllabus convention unless the question says otherwise — test a point like (0,0) if unsure which side is wanted.

Sequences

Pattern in differences	Type	nth term shape
Constant 1st difference	Linear	an + b
Constant 2nd difference	Quadratic	an² + bn + c
Constant, nonzero 3rd difference	Cubic	an³ + bn² + cn + d

Core asks for the nth term of linear, simple quadratic and simple cubic sequences (for example, 2, 5, 10, 17, … has nth term n² + 1). (Extended) Extended covers general quadratic and cubic sequences, and adds exponential sequences and simple combinations of these types.

Method: write out the sequence, then the differences between consecutive terms, then the differences of those (2nd differences), then — if the 2nd differences are still changing — the differences of those (3rd differences), to identify the type before finding the nth term. A cubic sequence is identified by a constant 3rd difference, not by a roughly constant ratio between 2nd differences; for n³, for example, the 2nd differences are 12, 18, 24, … (a changing ratio) while the 3rd differences are a constant 6.

Graphs

• Gradient of a straight line = (change in y) / (change in x), read from any two points on the line.
• Travel graphs: gradient of a distance–time graph = speed; a flat section = stationary; a negative gradient = returning toward the start.
• Reading vs sketching — keep separate:
  • Reading a given graph: use the axes and scale as drawn to find gradients, intercepts, or approximate solutions where the graph crosses a line.
  • Sketching (Core C2.11, linear/quadratic): show only the general shape and key features (intercepts, symmetry, direction) — no need for a table of values or exact scale. (Extended E2.11 adds cubic, reciprocal and exponential shapes, and requires turning points and asymptotes too.)
• Quadratic graph shape: a positive x² coefficient gives a U-shape (minimum); a negative x² coefficient gives an n-shape (maximum).

Extended-only: proportion, differentiation, functions

• Direct proportion: y ∝ x means y = kx. Inverse proportion: y ∝ 1/x means y = k/x. Find k from one known pair of values first, then use it to find the rest.
• Differentiation: for y = axⁿ, dy/dx = anxⁿ⁻¹. The gradient of a curve at a point equals the value of dy/dx at that point. A turning point occurs where dy/dx = 0; check the sign either side (or the second derivative) to tell a maximum from a minimum.
• Function notation: f(x) means “substitute x into the function f.” fg(x) means “apply g first, then apply f to the result” — work from the inside out. f⁻¹(x) undoes f(x); to find it algebraically, write y = f(x), swap x and y, then solve for y.

Common mistakes

• Writing (a + b)² = a² + b², forgetting the middle term 2ab.
• Sign errors when expanding brackets with a negative in front, e.g. −(x − 3) = −x + 3, not −x − 3.
• (Extended) Forgetting to reverse an inequality sign when multiplying/dividing by a negative number.
• Confusing “reading a graph” (use given axes) with “sketching a graph” (show shape and key features only).
• (Extended) In function composition, applying f and g in the wrong order — fg(x) means g first, then f.

Examiner report insight

• Incomplete factorisation: removing a numeric common factor is often only the first step – check what remains inside the bracket for a further structure, especially a difference of two squares (a^2 - b^2).
• Elimination sign errors: after scaling one equation to match coefficients, re-check whether the two equations should now be added or subtracted to eliminate the variable – this is where most marks are lost, not in the scaling itself.

Source: Cambridge International, 0580 Mathematics Principal Examiner Report, June 2024 series, Papers 11, 22 (verified 2026-09-02).

Self-test

1. (Extended) Factorise x² − 49.
2. Solve 3x − 7 = 11.
3. (Extended) Solve 4 − 2x ≥ 10, and represent it on a number line.
4. The first four terms of a sequence are 5, 8, 11, 14. Find the nth term.
5. (Extended) A straight line passes through (0, 3) and (4, 11). Find its gradient.

Answers: 1. (x + 7)(x − 7). 2. 3x = 18, x = 6. 3. −2x ≥ 6, x ≤ −3 (dividing by a negative reverses the sign) — filled circle at −3, shading to the left. 4. Constant difference of 3, so nth term = 3n + 2. 5. gradient = (11 − 3)/(4 − 0) = 8/4 = 2.