Cambridge O Level Additional Mathematics · Syllabus 4037 · Equations, Inequalities and Graphs
Substitution
What is Substitution?
Substitution replaces a repeated block of an equation with a single new variable, turning an equation that is not a quadratic into one that is. The new variable inherits a restriction from the block it replaces: a squared expression cannot be negative, an exponential is strictly positive, and the argument of a logarithm must be positive. Solving the quadratic gives values of the new variable, which must be filtered by that restriction and then converted back into values of the original variable before the question is answered.
This definition is part of the Equations, Inequalities and Graphs chapter in Cambridge O Level Additional Mathematics.
Substitution in context
Chapter 4 is about solving equations and inequalities that are not plain quadratics: modulus equations and inequalities, hidden quadratics solved by substitution, and cubic inequalities read from a graph. One idea runs through all of them: a rewrite is only safe while its conditions hold. Removing a modulus creates two branches, letting \(u=x^2\) means \(u\) can never be negative, and reading an inequality from a cubic only works if you compare the curve with the line you were actually given. Every candidate answer must be substituted back into the original statement before it counts as a solution.
A modulus measures distance from zero, so it never returns a negative value and it always offers two ways in. A substitution exposes a quadratic hiding inside a quartic, a logarithm or an exponential. A cubic drawn from its three factors turns an awkward inequality into interval reading. In every case the original equation or graph — not the convenient rewritten one — is the final authority.
Common mistakes with Substitution
- “I solved the quadratic and got \(u=1\) and \(u=4\), so the answers are 1 and 4.” Why it fails The question asked for \(x\), and \(u\) was never \(x\) — it was a temporary name for \(x^2\). Stopping at \(u\) reports the wrong quantity, and it also loses half the roots, because each positive \(u\) supplies two values of \(x\). Fix Write the substitution as a two-way arrow at the top of the working: \(u=x^2 \Leftrightarrow x=\pm\sqrt u\). The right-hand side is the return ticket, and having written it you are far less likely to leave without it.