Until about three years ago, I had literally no idea how to factorise nasty quadratics. I would turn straight to the quadratic formula, go bing bang boom and say ‘there, job done.’ This was a very effective short-cut – I got a long way with my ignorance – but I’m faintly embarrassed about it now. Especially since I was taught how to do it properly by two of my students (Kat and Cos – to whom I’m still grateful.)

Let’s start with a simple-ish quadratic expression, such as $x^2 -9x + 18$. The way to factorise is to find two numbers that multiply together to make 18 but add to make -9. Eighteen doesn’t have all that many factor pairs – (1, 18), (2, 9), (3,6) and their negative counterparts. The one we’re after is (-3, -6), which just drop into brackets with the $x$s to make $(x-3)(x-6)$. If you multiply that out, you get back to the original equation. So far so simple.

But when you have a number in front of the $x^2$, it’s a bit trickier. How about $2x^2 + 5x – 3$?

The method’s quite similar – except that the number you’re interested in is the first number times the last one ($2 \times -3 = -6$). The possible factors are (1, -6), (2, -3) and the negatives – so to get 5, I want (-1, 6). Here comes the clever bit!

You’re going to change the middle number (with the $x$) into the numbers you just found – so you rewrite the expression as $2x^2 – x + 6x – 3$.

Factorise the two halves of the equation to get $x(2x -1) + 3(2x -1)$, and glomp them together to get $(x + 3)(2x – 1)$. Finished!

One more example: $6x^2 + 5x – 6$.

You’re interested in $6 \times -6 = -36$, which has plenty of factors: (1,-36), (2,-18), (3,-12),(4,-9) and (6,-6) – and their negative friends. So (-4, 9) will give me 5 in the middle.

That leaves me with $6x^2 – 4x + 9x – 6$, which factorises as $2x(3x – 2) + 3(3x – 2)$, or $(2x + 3)(3x – 2)$.

Try it with these!

- $3x^2 – 8x – 3$
- $5x^2 – 4x – 1$
- $8x^2 – 10x + 3$
- $4x^2 + 7x + 3$

(If you get stuck, try using the formula and see where it’s gone pear-shaped!)

## Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008.
He lives with an espresso pot and nothing to prove.

## elvaton elkayz

This place is just great!….i can now manage to solve quadratic equations using factorisation method….thanks Colin.

## Colin

Thanks!

## Algy

Hmmmm; where are I going wrong here!?

8x^2 -10x +3

8*3 = 24

-12 and +2 factors

8x^2 -12x +2x +3

=> 4x(2x-3) = 8x^2 – 12x

??(2x-3) = 2x+3

## Colin

Be careful: -12 × 2 isn’t 24. Try -4 × -6.

## Adrian

When you say ‘Glomp them together’ how do you get x(2x−1)+3(2x−1) to become (x+3)(2x–1)? I am stuck on that bit.

BTW thanks great site – I’m enjoying reading about completing the square!

## Colin

Hi, Adrian, great question — and thanks for your kind words, glad you’re finding it useful!

Right, so you’re happy that something like $x\text{ Apples} + 3\text{ Apples} = (x+3)\text{ Apples}$? Similarly, $xz + 3z = (x+3)z$. In general, if you’ve got $x$ of something and $3$ of the same thing, you end up with $(x+3)$ of the thing. Here, the ‘thing’ is $(2x-1)$ — you have $x$ of it, added to $3$ of it, making $(x+3)$ of it!

Hope that helps 🙂

## Andy

Loved this method.

## Jonathan

3x^2 -8x-3

Result : 3x^2 -9x+x-3

3x(x-3)+1(x-3)

(x-3)[3x+1]

(3x+1)(x-3)

## Colin

Great!

## trinankur saha

It is helpful

## EMKAY MANNE

That really helps but please help me with the following equation:

6m2-2mn-3mp+np (The 2 after 6m is supposed to be squared.)

## Colin

Hi, Emkay!

My approach to that would be pretty similar, actually: treat $m$ as your $x$, and everything else as numbers.

You’ve then got $6m^2 + (-2n – 3p)m + np$, and you want a pair of factors that multiply together to make $6np$ but add together to make $-2n-3p$. Can you take it from there?

## Sakshi

So what will be the factors of 2x^2+3x-90?

## Colin

Hi, Sakshi,

What have you tried? I’d start by listing factors of -180 and seeing which pair has a difference of 3.

C

## John

3x^2-x-1

3*-1=-3

this is a prime number what do I do then?

## Colin

Hi, John,

In this context, there’s nothing special about prime numbers: you look at the possible factors of -3: either -1 and 3 (which sum to 2) or -3 and 1 (which sum to -2). There is no (integer) pair of factors that sums to -1, so this quadratic doesn’t factorise. Instead, you can solve it by completing the square or using the formula.

## Rotimi

How do I go about x³+8

And x²-4

## Colin

The first of those is not a quadratic – you would probably need to spot a solution (there’s one obvious real number that works) and then use long division or similar to find the other factor.

If you don’t spot the pattern for $x^2 – 4$, you can turn it into $x^2 + 0x – 4$ and use the usual factorising techniques.

## Sakshi

I got that sir

It is 15 & 12

## Colin

OK, so now follow the method. You’ve got -12x and 15x in the middle, making it $2x^2 – 12x + 15x – 90$. What do you get if you factorise the two halves?

## Sakshi

After solving it i got (2x+15)(x-6)

## Andrian

I got this…

6x^2-x-6x+2

x(6x-1)+2(-3x+1)…

## Colin

For $6x^2 -7x + 2$?

You would need a pair of numbers that added to -7 (which -1 and -6 do) but multiply to make $6\times 2=12$, which -1 and -6 don’t. See if you can make it work with another pair.

## maeva

Hello And for x²-2x-5 please cuz i’m stuck

## Colin

I’m afraid that one doesn’t factorise. If that’s equal to zero, you can solve it by completing the square or applying the formula – I get $x^2-2x-5 \equiv (x-1)^2 – 6$, so its zeros will be at $x=1\pm \sqrt{6}$.

## Kate

How do i solve this?

3a^2-4a-15=0

## Colin

Exactly the same way! You need to find two numbers that multiply to make -45, but add to make -4. I’d say 5 and -9 fit the bill nicely.

Then $3a^2 – 4a – 15 = 3a^2 – 9a + 5a – 15 = 3a(a-3) + 5(a-3) = (3a+5)(a-3)$.

That gives solutions of $a=-\frac{5}{3}$ and $a=3$.