What is a quadratic equation?
A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. It usually has two solutions (roots), but may have one repeated root or none (real). Three methods are used: factorising, the quadratic formula, and (less often at this level) completing the square.
Key idea
The quadratic formula: x = (−b ± √(b² − 4ac)) / (2a). The discriminant b² − 4ac tells you the number of real roots: positive → 2, zero → 1, negative → 0.
Factorising
If the product of two brackets is zero, at least one bracket must be zero (the zero-product rule). Set each factor equal to 0 and solve.
Worked example
Solve x² + 5x + 6 = 0.
Factorise: two numbers with product 6 and sum 5 are 2 and 3.
(x + 2)(x + 3) = 0.
So x + 2 = 0 or x + 3 = 0, giving x = −2 or x = −3.
Check: (−2)² + 5(−2) + 6 = 4 − 10 + 6 = 0. ✓
Remember
- x² = 49 has two answers: x = +7 and x = −7.
- Always write "= 0" before factorising.
- If it will not factorise neatly, use the formula.