Algebraic expressions, equations, systems and identities
Seeing structure before expanding
Algebraic competence means recognising a difference of squares, a perfect-square trinomial, a sum or difference of cubes, and grouping patterns before expanding blindly. The identities (a ± b)², a² − b², a³ ± b³ are shortcuts, not decoration.
For quadratics, know three routes: factoring, the discriminant formula D = b² − 4ac, and Vieta's relations x₁ + x₂ = −b/a, x₁x₂ = c/a. Vieta answers sum-and-product questions instantly and reveals sign patterns without solving.
Systems: substitution when one equation is linear in a variable; elimination when coefficients align; and the symmetric substitution u = x + y, v = xy for systems symmetric in x and y.
Traps. Squaring both sides can create extraneous roots - always verify in the original equation. Dividing by an expression containing the unknown destroys solutions; factor instead. For equations with roots, logarithms or fractions, the domain must be checked first. In identity questions, "for all x" means coefficients of like powers must match.
Worked example. Solve x + √x − 6 = 0. Put t = √x ≥ 0. Then t² + t − 6 = 0, so (t + 3)(t − 2) = 0, giving t = 2 (t = −3 is rejected). Hence x = 4, and checking: 4 + 2 − 6 = 0. ✓
Rule of thumb: substitution turns a disguised quadratic into a plain one, but the substitution's own range (t ≥ 0, t > 0, |t| ≤ 1) must be enforced at the end.