The idea

Solving quadratics by factorising

Why a product being zero forces a factor to be zero, why the equation must equal zero BEFORE factorising helps, and how to factorise with a leading coefficient and the difference of two squares.

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Algebra · Quadratic equations

Solving quadratics by factorising

Why a product being zero forces a factor to be zero, why the equation must equal zero BEFORE factorising helps, and how to factorise with a leading coefficient and the difference of two squares.

Why it works

Zero forces its factors

The whole method rests on one special fact about zero. If two numbers multiply to zero, one of them must be zero — no other product forces its factors like that. So from

(x - 2)(x + 7) = 0

we can conclude

x - 2 = 0

x + 7 = 0

, giving

x = 2

x = -7

. Notice the sign flip: the bracket

(x + 7)

dies when

x

-7

, not

7

— read each root by asking "what kills this bracket?"

Only when one side IS zero

(x - 2)(x + 7) = 10

tells you almost nothing — 10 factorises as

2 \times 5

1 \times 10

-2 \times -5

, endlessly. Only zero forces its factors. Faced with

x(x + 2) = 24

, don't chase factor pairs of 24: expand, drag everything to one side, and then factorise:

x^2 + 2x - 24 = 0 \quad\Rightarrow\quad (x + 6)(x - 4) = 0 \quad\Rightarrow\quad x = -6 \text{ or } 4

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