This mini-lesson covers Algebra & Graphs for Cambridge IGCSE Mathematics (0580): algebraic manipulation, equations, simultaneous & quadratic equations, inequalities, sequences, functions and graphs of functions (Extended content included).
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Expand brackets carefully, then collect like terms. Factorise by taking out the highest common factor, or expand a double bracket with a grid / FOIL.
Sign care: (x − 4)(x + 4) = x² − 16, a difference of two squares — the middle terms cancel.
Do the same to both sides to keep the balance. Undo operations in reverse order (inverse of + is −, of × is ÷).
Solve 3x + 4 = 19.
3x = 15 → x = 5
Two equations, two unknowns. Eliminate a variable by adding or subtracting (make coefficients match), or substitute.
x + y = 10 and x − y = 4. Add: 2x = 14 → x = 7. Then y = 3.
Solution: x = 7, y = 3 (check: 7+3 = 10 ✓)
Solve x² + bx + c = 0 by factorising (find two numbers that multiply to c and add to b), or use the quadratic formula.
x² − 7x + 12 = 0 → (x − 3)(x − 4) = 0 → x = 3 or x = 4
Solve like an equation, but reverse the inequality sign whenever you multiply or divide by a negative number.
Solve 2x + 1 < 11 → 2x < 10 → x < 5.
Watch the flip: −x < 3 becomes x > −3 (divide by −1, flip the sign).
For a linear (arithmetic) sequence the nth term is dn + (a − d), where d is the common difference and a is the first term.
5, 8, 11, 14… d = 3, first term 5, so nth term = 3n + 2.
Check n = 1: 3(1) + 2 = 5 ✓
Tap a sequence, then its correct nth-term rule.
A function f(x) is a rule; f(3) means substitute x = 3. The inverse f⁻¹(x) undoes it. Recognise graph shapes: linear (straight), quadratic (parabola), cubic, reciprocal (y = 1/x) and exponential.
f(x) = 2x + 5. Then f(4) = 2(4) + 5 = 13.
Sort each equation by its graph shape. Tap it, then its box.
Manipulation: expand double brackets; factorise; difference of two squares.
Equations: keep the balance; simultaneous by elimination/substitution.
Quadratics: factorise or use x = [−b ± √(b²−4ac)]/2a.
Inequalities: flip the sign when ÷ or × by a negative.
Sequences: linear nth term dn + (a − d).
Functions & graphs: f(x) substitution; recognise linear/quadratic/cubic/reciprocal/exponential shapes.
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