Cambridge IGCSE Mathematics · Syllabus 0580 · Algebra and Graphs
Equations
What is Equations?
A statement that two expressions are equal, true only for particular values of the unknown, which solving is the process of finding. Linear equations are solved by inverse operations, fractional equations by clearing denominators after recording their excluded values, simultaneous linear equations by elimination or substitution, and quadratic equations by factorisation, by completing the square or by the quadratic formula. Changing the subject rearranges a formula so that a different letter stands alone, and any solution that makes an original denominator zero must be rejected. Linear equations, simultaneous linear equations and simple subject changes are required on both routes; fractional equations, mixed linear and non-linear pairs, quadratic equations and harder subject changes are Extended requirements.
This definition is part of the Algebra and Graphs chapter in Cambridge IGCSE Mathematics.
Equations in context
A drawn graph is a solving machine. Once the curve \(y=f(x)\) is on the paper, you can read off the solution to \(f(x)=0\) (where it crosses the \(x\)-axis), to \(f(x)=k\) (where it crosses the horizontal line \(y=k\)), and to a whole family of rearranged equations, by adding the right straight line. That last idea is the most powerful thing in this section, and the least obvious — and it is required on both routes.
Examiner tips on Equations
- The List of formulas. A List of formulas is printed in the question paper. For this topic the one that matters is the quadratic formula: for \(ax^2+bx+c=0\) with \(a\ne0\), \[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.\] That is an Extended requirement (E2.5) — quadratic equations are not part of the Core course — so it matters on Papers 2 and 4 only. You do not need to memorise it, but you do need to be fluent enough with it that reading it off the sheet costs seconds, not minutes. Nothing else in Topic 2 is given: the difference of two squares, the completed-square procedure, the index laws, the \(n\)th-term methods and the derivative of \(ax^n\) are all recall.