Solving by factorising

Factorising is the quickest way to solve a quadratic, when it works. It rests on one simple fact.

Why brackets give the answers

If two numbers multiply to give 0, at least one of them must be 0. There’s no other way to get 0 from a multiplication.

So if $(x - 3)(x + 2) = 0$, then either $x - 3 = 0$ or $x + 2 = 0$. That gives $x = 3$ or $x = -2$.

Finding the brackets

Multiply out $(x + 3)(x + 4)$ and you get $x^2 + 7x + 12$. Notice that $3 \times 4 = 12$ and $3 + 4 = 7$. Factorising runs this backwards:

• To factorise $x^2 + bx + c$, find two numbers that multiply to $c$ and add to $b$.
• When there’s a number in front of $x^2$, as in $ax^2 + bx + c$, find two numbers that multiply to $ac$ and add to $b$. Use them to split the middle term into two, then take out common factors in pairs.

Try it. Pick the right pair; a wrong pick tells you which part failed.

A past question, step by step

Going backwards: from roots to the equation

Exam questions often give you the roots and ask for the equation. Turn each root into a bracket, then multiply out.

• A root $x = 3$ comes from the bracket $(x - 3)$.
• A fraction root such as $x = -\frac35$: clear the fraction first. $5x = -3$, so $5x + 3 = 0$, and the bracket is $(5x + 3)$.

So the equation with roots $4$ and $-\frac35$ is $(x - 4)(5x + 3) = 0$, which is $5x^2 - 17x - 12 = 0$.

The sum and product of the roots

If $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$, you can find their sum and product without solving:

For example, for $2x^2 - 7x + 3 = 0$ the roots add to $\frac72$ and multiply to $\frac32$. Other expressions are built from these two: $\frac1\alpha + \frac1\beta = \frac{\alpha + \beta}{\alpha\beta}$.

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