Solving Quadratic Equations by Factorisation

Why factorisation solves the equation

If two numbers multiply to make zero, at least one of them must be zero. This is the zero-product rule:

$AB=0 \Rightarrow A=0 \text{ or } B=0$

So once a quadratic is written as two factors multiplied together, each factor gives a possible solution.

1. Start with a factorised equation

Example 1

Solve $(x-3)(x+5)=0$.

$x-3=0$ or $x+5=0$

$x=3$ or $x=-5$

2. Factorise a monic quadratic

Example 2

Solve $x^2+7x+12=0$.

$x^2+7x+12=(x+3)(x+4)$

$(x+3)(x+4)=0$

$x=-3$ or $x=-4$

Example 3

Solve $x^2-x-12=0$.

$x^2-x-12=(x-4)(x+3)$

$x=4$ or $x=-3$

3. Rearrange first

The zero-product rule only works when the product equals zero.

Example 4

Solve $x^2+5x=14$.

$x^2+5x-14=0$

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

$x=-7$ or $x=2$

4. Recognise special quadratics

Example 5: difference of two squares

Solve $x^2-49=0$.

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

$x=-7$ or $x=7$

Example 6: repeated root

Solve $x^2-10x+25=0$.

$(x-5)^2=0$

$x=5$

The same factor appears twice, so there is one repeated solution.

5. Take out a common factor

Example 7

Solve $3x^2-15x+18=0$.

$3(x^2-5x+6)=0$

$3(x-2)(x-3)=0$

$x=2$ or $x=3$

Common mistakes

Factorising correctly but stopping before finding $x$.
Using the zero-product rule when the other side is not zero.
Finding one solution and forgetting the other.
Changing a sign when solving a linear factor, such as $x+4=0$.