A quadratic equation is of the form ax² + bx + c = 0, where a ≠ 0 and a, b, c are real numbers. Examples: 2x² – 3x + 1 = 0, x² – 5 = 0. It can have 0, 1, or 2 real roots.

Split the middle term to factorise ax² + bx + c = 0. Find two numbers whose sum = b and product = ac, then split and group. Example: x² – 5x + 6 = 0 → (x–2)(x–3) = 0 → x = 2 or x = 3.

If ax² + bx + c = 0, then x = [–b ± √(b²–4ac)] / 2a. The discriminant D = b²–4ac determines nature of roots: D > 0 → two distinct real roots, D = 0 → two equal real roots, D < 0 → no real roots.

To solve ax² + bx + c = 0 by completing the square: divide by a, shift constant, add (b/2a)² to both sides, take square root, solve for x. This method is the basis for deriving the quadratic formula.

Many real-life problems lead to quadratic equations. Common types: (1) Number problems: "A number and its reciprocal sum to 10/3 → x + 1/x = 10/3 → 3x²−10x+3=0"; (2) Speed-distance problems: reduced speed increases time; (3) Area problems: dimensions of a rectangle or plot; (4) Age problems. Strategy: let the unknown be x, form the equation carefully, solve, and reject extraneous or negative solutions that don't fit the real-world context. Always verify both roots.