Overview
Quadratics have the form y = ax² + bx + c. Know completing the square and the discriminant.
Completing the square
Write as a(x + p)² + q. This instantly gives the turning point at (−p, q) — a minimum if a > 0, a maximum if a < 0.
The discriminant
For ax² + bx + c = 0, the discriminant is b² − 4ac:
- > 0 → two distinct real roots
- = 0 → one repeated root (the line is a tangent)
- < 0 → no real roots
The quadratic formula
x = [−b ± √(b² − 4ac)] ÷ 2a.
Worked example
x² − 6x + 5: complete the square → (x − 3)² − 4. Minimum point at (3, −4); roots at x = 1 and x = 5.
Common mistakes
- Forgetting the sign change: a(x + p)² + q has its turning point at x = −p.
Exam tips
- "Line is a tangent to the curve" → set the discriminant = 0.