Factorization
Factorisation is the fastest way to solve a quadratic when the roots are rational, and a sharp CAT student tries it before reaching for the formula. The method (when a = 1): find two numbers that multiply to c and add to b, then split the middle term. For x² − 7x + 12, the pair is −3 and −4 (product 12, sum −7), giving (x − 3)(x − 4) = 0. When a ≠ 1, find two numbers whose product is a·c and whose sum is b, split bx accordingly, then group. By the zero-product rule, if a product equals 0 then one factor is 0, so each factor set to zero gives a root. A quick CAT check: factorisation only works neatly when D = b² − 4ac is a perfect square; if it is not, the roots are irrational and the formula is faster than guessing. Always confirm the coefficient signs of the constant term to fix the signs of the factors.
✅ 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 |