A quadratic is an expression with an x² term in it, like x² + 7x + 12. They describe the shape of a thrown ball, the path of water from a fountain, and the area of a square whose side you don't yet know — anything where a quantity depends on the square of something. At GCSE the key skill is factorising them into two brackets.
What "factorising" undoes
Expanding (x + 3)(x + 4) with the grid or FOIL method gives x² + 4x + 3x + 12 = x² + 7x + 12. Factorising is that process in reverse — you start with x² + 7x + 12 and work back to the two brackets. Notice where the numbers came from: the 12 is 3 × 4, and the 7 is 3 + 4. That is the whole secret.
Factorising x² + bx + c
Find two numbers that multiply to give c (the last number) and add to give b (the middle number). Those two numbers go into the brackets.
Signs are a helpful clue. Think of the pair of numbers you need:
| Sign of c | Sign of b | The two numbers are… | Example |
|---|---|---|---|
| + | + | both positive | x² + 7x + 12 → (x+3)(x+4) |
| + | − | both negative | x² − 7x + 12 → (x−3)(x−4) |
| − | either | one positive, one negative | x² + x − 12 → (x+4)(x−3) |
When c is negative, the two numbers have opposite signs, and the bigger one takes the sign of b. That single rule clears up most sign muddles.
Solving a quadratic equation
To solve "= 0", factorise first, then use this powerful idea:
Analogy: if you multiply two numbers and get zero, at least one of them was zero — there's no other way. Nothing times nothing-but-zero gives zero. That's why each bracket gets its own turn at being zero.
Difference of two squares — the special case
Sometimes there is no middle term, just a square number being subtracted, like x² − 9. Here the two numbers multiply to −9 and add to 0, so they must be +3 and −3.