Maximum & Minimum
A quadratic ax² + bx + c traces a parabola, and its turning point is the key to fast CAT optimisation — no calculus required. The vertex sits at x = −b/2a, and the extreme value of the expression there is −D/4a (equivalently c − b²/4a). If a > 0 the parabola opens upward, so this value is a MINIMUM; if a < 0 it opens downward, so it is a MAXIMUM. That single rule cracks a whole class of questions: maximum area for a fixed perimeter, maximum product of two numbers with a fixed sum, the least value of an expression. A useful corollary for "two numbers with a fixed sum S": their product is largest when they are equal (each S/2), giving S²/4 — directly the vertex result. Watch the sign of a first: the most common CAT error is reporting a minimum for a downward parabola (a < 0), which actually has no minimum (it falls to −∞).
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
Solving & nature of roots
| Quadratic formula | x = [−b ± √(b² − 4ac)] / 2a |
|---|---|
| Discriminant | D = b² − 4ac |
| Two distinct real roots | D > 0 |
| Equal (repeated) real roots | D = 0 ⇒ x = −b/2a |
| No real roots (complex pair) | D < 0 |
Roots, building & extremes
| Sum of roots | α + β = −b/a |
|---|---|
| Product of roots | αβ = c/a |
| Equation from roots | x² − (α + β)x + αβ = 0 |
| Vertex (turning point) | x = −b/2a, value = −D/4a |
| Min if a > 0, Max if a < 0 | extreme value = c − b²/4a = −D/4a |