Solving by factorisation

A root is a number that makes the equation true. Put x = 1 into 2x² – 3x + 1 = 0: 2 – 3 + 1 = 0 ✓, so 1 is a root. The roots of the equation are the same as the zeroes of the polynomial, so a quadratic equation has at most two roots. Sometimes it has two, sometimes one repeated root, and sometimes none.

The key fact: if two things multiply to give 0, at least one of them must be 0. So if (2x – 3)(x – 1) = 0, then 2x – 3 = 0 or x – 1 = 0.

Splitting the middle term. For 2x² – 5x + 3: find two numbers that multiply to a × c = 6 and add to b = –5. They are –2 and –3.

2x² – 2x – 3x + 3 = 2x(x – 1) – 3(x – 1) = (2x – 3)(x – 1), so x = 3/2 or x = 1.

If you can't see the two numbers, break a × c into primes and regroup. For the prayer hall, 2x² + x – 300: a × c = –600 = –(2³ × 3 × 5²). Regroup as 25 and –24, which add to 1. You get (x – 12)(2x + 25) = 0.

Three traps:

• x² – 16 = 0 has two roots, 4 and –4.
• In x² – 4x = 0, never divide by x: you would lose the root 0. Take x out instead: x(x – 4) = 0, so x = 0 or 4.
• A repeated factor gives one root twice: 3x² – 2√6x + 2 = (√3x – √2)², so both roots are √(2/3).

Throw out roots that make no sense. The hall's breadth is 12 m or –12.5 m. A breadth can't be negative, so it is 12 m, and the length is 25 m. Say why you reject a root: a length, a speed, a digit or a number of people can't be negative.