Lesson 12: Factorising x^{2}+bx+c to double brackets

Introduction

In Lesson 11, you expanded double brackets, each bracket containing an
\(~x~\) term and a constant (number) term. Your answer was an expression with
an \(~x^2~\) term, an \(~x~\) term and a constant term. If we want to show off,
we could write this expression as
\(~ax^2+bx+c~\), where \(~a~\), \(~b~\) and \(~c~\) can be any numbers
(positive or negative). This is called the general form of a
quadratic expression
(quadratic just means that the highest power of \(~x~\) is 2).

This lesson, you're going to do the exact opposite. You're going to take a
quadratic expression and put it back into double brackets. Since the opposite
of expanding is factorising, you're going to be doing
quadratic factorisation.

Because having a number in front of the \(~x^2~\) term
(a coefficient of \(~x^2~\) greater than 1), makes the method more complicated,
we'll leave that until Lesson 13. This is important stuff and we want to take
the time to get it right.

SECTION A

OK, let's say we want to factorise \(~x^2+7x+10~\) (put it back into double
brackets).

Step 1

Find two beautiful numbers that
multiply together to make \(~c~\)
and add together to make \(~b~\).
Remember \(~c~\) is the constant term (in this case 10) and \(~b~\) is the
coefficient of the \(~x~\) term (in this case 7). So we want two numbers that
multiply together to make 10 and add together to make 7.

Well, the numbers that multiply together to make 10 could be 1 and 10 but
they could also be 2 and 5.
Ah, but only the 2 and 5 add together to make 7. So the beautiful numbers we're
looking for are 2 and 5!

Step 2

Every answer starts like this: \(~(x~~~~~~~~)(x~~~~~~~~)~\). Just put your
beautiful numbers in the spaces provided and you're done!

Answer: \(~(x+2)(x+5)~\)

Don't believe me? Just expand your answer back out and you'll get
\(~x^2+7x+10~\). In fact, you should always check factorisation by
expanding your answer back out.

That's it for now! But please work through the following examples carefully.
Then you'll really know what you're doing and you can push on with confidence.

Example 1

Factorise \(~x^2+6x+5~\)

Two beautiful numbers multiply to make 5 and add to make 6.

\(~1\times 5=5~\) and \(~1+5=6~\)

Our beautiful numbers are 1 and 5.

Answer: \(~(x+1)(x+5)~\)

In case you're wondering, I could have written \(~(x+5)(x+1)~\) instead.
It doesn't matter which bracket is put first.

Example 2

Factorise \(~x^2+10x+24~\)

Two beautiful numbers multiply to make 24 and add to make 10.

\(~1\times 24=24~\)

\(~2\times 12=24~\)

\(~3\times 8=24~\)

\(~4\times 6=24~\)

Ah, but of these, only \(~4+6=10~\)

So our beautiful numbers are 4 and 6.

Answer: \(~(x+4)(x+6)~\)

There are plenty of practice questions and for good reason!

Practise to master

SECTION A

Factorise the following expressions.

01) \(~x^2+8x+12~\)

Two beautiful numbers \(\times\) to make 12 and \(+\) to make 8
\(2\times 6=12\) and \(2+6=8\)
Our beautiful numbers are 2 and 6
Answer: \((x+2)(x+6)\)
I'm not going to write down all the pairs of numbers that \(\times\) to make \(c\) but you probably should for a while. Often there is only one pair, which is nice!

02) \(~x^2+3x+2~\)

Two beautiful numbers \(\times\) to make 2 and \(+\) to make 3
\(1\times 2=2\) and \(1+2=3\)
Our beautiful numbers are 1 and 2
Answer: \((x+1)(x+2)\)

03) \(~x^2+7x+6~\)

Two beautiful numbers \(\times\) to make 6 and \(+\) to make 7
\(1\times 6=6\) and \(1+6=7\)
Our beautiful numbers are 1 and 6
Answer: \((x+1)(x+6)\)

04) \(~x^2+4x+3~\)

Two beautiful numbers \(\times\) to make 3 and \(+\) to make 4
\(1\times 3=3\) and \(1+3=4\)
Our beautiful numbers are 1 and 3
Answer: \((x+1)(x+3)\)

05) \(~x^2+9x+20~\)

Two beautiful numbers \(\times\) to make 20 and \(+\) to make 9
\(4\times 5=20\) and \(4+5=9\)
Our beautiful numbers are 4 and 5
Answer: \((x+4)(x+5)\)

06) \(~x^2+8x+15~\)

Two beautiful numbers \(\times\) to make 15 and \(+\) to make 8
\(3\times 5=15\) and \(3+5=8\)
Our beautiful numbers are 3 and 5
Answer: \((x+3)(x+5)\)

07) \(~x^2+2x+1~\)

Two beautiful numbers \(\times\) to make 1 and \(+\) to make 2
\(1\times 1=1\) and \(1+1=2\)
Our beautiful numbers are 1 and 1
Answer: \((x+1)(x+1)\) or even better \((x+1)^2\)

08) \(~x^2+5x+6~\)

Two beautiful numbers \(\times\) to make 6 and \(+\) to make 5
\(2\times 3=6\) and \(2+3=5\)
Our beautiful numbers are 2 and 3
Answer: \((x+2)(x+3)\)

09) \(~x^2+10x+21~\)

Two beautiful numbers \(\times\) to make 21 and \(+\) to make 10
\(3\times 7=21\) and \(3+7=10\)
Our beautiful numbers are 3 and 7
Answer: \((x+3)(x+7)\)

10) \(~x^2+13x+40~\)

Two beautiful numbers \(\times\) to make 40 and \(+\) to make 13
\(5\times 8=40\) and \(5+8=13\)
Our beautiful numbers are 5 and 8
Answer: \((x+5)(x+8)\)

11) \(~x^2+11x+30~\)

Two beautiful numbers \(\times\) to make 30 and \(+\) to make 11
\(5\times 6=30\) and \(5+6=11\)
Our beautiful numbers are 5 and 6
Answer: \((x+5)(x+6)\)

12) \(~x^2+10x+25~\)

Two beautiful numbers \(\times\) to make 25 and \(+\) to make 10
\(5\times 5=25\) and \(5+5=10\)
Our beautiful numbers are 5 and 5
Answer: \((x+5)(x+5)\) or even better \((x+5)^2\)

SECTION B

If you want to take quadratic factorisation to the next level, you need to
be fluent in multiplying and adding with negative numbers!

Example 1

Factorise \(~x^2+3x-4~\)

Two beautiful numbers multiply to make \(-\)4 and add to make 3.

\(~1\times -4=-4~\)

\(~-1\times 4=-4~\)

\(~2\times -2=-4~\)

But of these, only \(~-1+4=3~\)

So our beautiful numbers are \(-\)1 and 4.

Answer: \(~(x-1)(x+4)~\)

Beware!

The sign to the left of a term always belongs to that term. So the
constant term in example 1 is NOT 4, it's \(-\)4. The coefficient of \(x\) is
just 3 as this is the same as +3, of course!

Example 2

Factorise \(~x^2-3x-18~\)

Two beautiful numbers multiply to make \(-\)18 and add to make \(-\)3.

\(~1\times -18=-18~\)

\(~-1\times 18=-18~\)

\(~2\times -9=-18~\)

\(~-2\times 9=-18~\)

\(~3\times -6=-18~\)

\(~-3\times 6=-18~\)

But of these, only \(~3+(-6)=-3~\)

So our beautiful numbers are 3 and \(-\)6.

Answer: \(~(x+3)(x-6)~\)

Example 3

Factorise \(~x^2-4x+3~\)

Two beautiful numbers multiply to make 3 and add to make \(-\)4.

This is the one that catches people out - minus \(\times\) minus = plus!

\(~1\times 3=3~\)

\(~-1\times -3=3~\)

But of these, only \(~-1+(-3)=-4~\)

So our beautiful numbers are \(-\)1 and \(-\)3.

Answer: \(~(x-1)(x-3)~\)

A few final words of advice!

You don't have to write down all the multiplication possibilities. You may
start, realise part way through you've found what you're looking for then stop.
You might eventually do this all in your head even with negatives.

Don't be caught out by a positive \(b\) and a negative \(c\) as in the last
example: negative \(\times\) negative = positive!

All you need now is lots of practice!

Practise to master

SECTION B

Factorise the following expressions.

01) \(~x^2+5x-6~\)

Two beautiful numbers \(\times\) to make \(-\)6 and \(+\) to make 5
\(-1\times 6=-6\) and \(-1+6=5\)
Our beautiful numbers are \(-\)1 and 6
Answer: \((x-1)(x+6)\)

02) \(~x^2-x-6~\)

Two beautiful numbers \(\times\) to make \(-\)6 and \(+\) to make \(-\)1
\(2\times -3=-6\) and \(2+(-3)=-1\)
Our beautiful numbers are 2 and \(-\)3
Answer: \((x+2)(x-3)\)

03) \(~x^2-6x+5~\)

Two beautiful numbers \(\times\) to make 5 and \(+\) to make \(-\)6
\(-1\times -5=5\) and \(-1+(-5)=-6\)
Our beautiful numbers are \(-\)1 and \(-\)5
Answer: \((x-1)(x-5)\)

04) \(~x^2+4x-12~\)

Two beautiful numbers \(\times\) to make \(-\)12 and \(+\) to make 4
\(-2\times 6=-12\) and \(-2+6=4\)
Our beautiful numbers are \(-\)2 and 6
Answer: \((x-2)(x+6)\)

05) \(~x^2-x-12~\)

Two beautiful numbers \(\times\) to make \(-\)12 and \(+\) to make \(-\)1
\(3\times -4=-12\) and \(3+(-4)=-1\)
Our beautiful numbers are 3 and \(-\)4
Answer: \((x+3)(x-4)\)

06) \(~x^2-2x+1~\)

Two beautiful numbers \(\times\) to make 1 and \(+\) to make \(-\)2
\(-1\times -1=1\) and \(-1+(-1)=-2\)
Our beautiful numbers are \(-\)1 and \(-\)1
Answer: \((x-1)(x-1)\) or even better \((x-1)^2\)

07) \(~x^2+3x-10~\)

Two beautiful numbers \(\times\) to make \(-\)10 and \(+\) to make 3
\(-2\times 5=-10\) and \(-2+5=3\)
Our beautiful numbers are \(-\)2 and 5
Answer: \((x-2)(x+5)\)

08) \(~x^2-x-20~\)

Two beautiful numbers \(\times\) to make \(-\)20 and \(+\) to make \(-\)1
\(4\times -5=-20\) and \(4+(-5)=-1\)
Our beautiful numbers are 4 and \(-\)5
Answer: \((x+4)(x-5)\)

09) \(~x^2-7x+12~\)

Two beautiful numbers \(\times\) to make 12 and \(+\) to make \(-\)7
\(-3\times -4=12\) and \(-3+(-4)=-7\)
Our beautiful numbers are \(-\)3 and \(-\)4
Answer: \((x-3)(x-4)\)

10) \(~x^2+2x-24~\)

Two beautiful numbers \(\times\) to make \(-\)24 and \(+\) to make 2
\(-4\times 6=-24\) and \(-4+6=2\)
Our beautiful numbers are \(-\)4 and 6
Answer: \((x-4)(x+6)\)

11) \(~x^2-2x-3~\)

Two beautiful numbers \(\times\) to make \(-\)3 and \(+\) to make \(-\)2
\(1\times -3=-3\) and \(1+(-3)=-2\)
Our beautiful numbers are 1 and \(-\)3
Answer: \((x+1)(x-3)\)

12) \(~x^2-9x+20~\)

Two beautiful numbers \(\times\) to make 20 and \(+\) to make \(-\)9
\(-4\times -5=20\) and \(-4+(-5)=-9\)
Our beautiful numbers are \(-\)4 and \(-\)5
Answer: \((x-4)(x-5)\)

13) \(~x^2+x-6~\)

Two beautiful numbers \(\times\) to make \(-\)6 and \(+\) to make 1
\(-2\times 3=-6\) and \(-2+3=1\)
Our beautiful numbers are \(-\)2 and 3
Answer: \((x-2)(x+3)\)

14) \(~x^2-2x-8~\)

Two beautiful numbers \(\times\) to make \(-\)8 and \(+\) to make \(-\)2
\(2\times -4=-8\) and \(2+(-4)=-2\)
Our beautiful numbers are 2 and \(-\)4
Answer: \((x+2)(x-4)\)

15) \(~x^2-3x+2~\)

Two beautiful numbers \(\times\) to make 2 and \(+\) to make \(-\)3
\(-1\times -2=2\) and \(-1+(-2)=-3\)
Our beautiful numbers are \(-\)1 and \(-\)2
Answer: \((x-1)(x-2)\)

16) \(~x^2+2x-15~\)

Two beautiful numbers \(\times\) to make \(-\)15 and \(+\) to make 2
\(-3\times 5=-15\) and \(-3+5=2\)
Our beautiful numbers are \(-\)3 and 5
Answer: \((x-3)(x+5)\)

17) \(~x^2-x-30~\)

Two beautiful numbers \(\times\) to make \(-\)30 and \(+\) to make \(-\)1
\(5\times -6=-30\) and \(5+(-6)=-1\)
Our beautiful numbers are 5 and \(-\)6
Answer: \((x+5)(x-6)\)

18) \(~x^2-12x+36~\)

Two beautiful numbers \(\times\) to make 36 and \(+\) to make \(-\)12
\(-6\times -6=36\) and \(-6+(-6)=-12\)
Our beautiful numbers are \(-\)6 and \(-\)6
Answer: \((x-6)(x-6)\) or even better \((x-6)^2\)

19) \(~x^2+x-2~\)

Two beautiful numbers \(\times\) to make \(-\)2 and \(+\) to make 1
\(-1\times 2=-2\) and \(-1+2=1\)
Our beautiful numbers are \(-\)1 and 2
Answer: \((x-1)(x+2)\)

20) \(~x^2-4x-5~\)

Two beautiful numbers \(\times\) to make \(-\)5 and \(+\) to make \(-\)4
\(1\times -5=-5\) and \(1+(-5)=-4\)
Our beautiful numbers are 1 and \(-\)5
Answer: \((x+1)(x-5)\)

21) \(~x^2-9x+14~\)

Two beautiful numbers \(\times\) to make 14 and \(+\) to make \(-\)9
\(-2\times -7=14\) and \(-2+(-7)=-9\)
Our beautiful numbers are \(-\)2 and \(-\)7
Answer: \((x-2)(x-7)\)

22) \(~x^2+x-30~\)

Two beautiful numbers \(\times\) to make \(-\)30 and \(+\) to make 1
\(-5\times 6=-30\) and \(-5+6=1\)
Our beautiful numbers are \(-\)5 and 6
Answer: \((x-5)(x+6)\)

23) \(~x^2-3x-4~\)

Two beautiful numbers \(\times\) to make \(-\)4 and \(+\) to make \(-\)3
\(1\times -4=-4\) and \(1+(-4)=-3\)
Our beautiful numbers are 1 and \(-\)4
Answer: \((x+1)(x-4)\)

24) \(~x^2-11x+30~\)

Two beautiful numbers \(\times\) to make 30 and \(+\) to make \(-\)11
\(-5\times -6=30\) and \(-5+(-6)=-11\)
Our beautiful numbers are \(-\)5 and \(-\)6
Answer: \((x-5)(x-6)\)

25) \(~x^2+3x-18~\)

Two beautiful numbers \(\times\) to make \(-\)18 and \(+\) to make 3
\(-3\times 6=-18\) and \(-3+6=3\)
Our beautiful numbers are \(-\)3 and 6
Answer: \((x-3)(x+6)\)

26) \(~x^2-2x-24~\)

Two beautiful numbers \(\times\) to make \(-\)24 and \(+\) to make \(-\)2
\(4\times -6=-24\) and \(4+(-6)=-2\)
Our beautiful numbers are 4 and \(-\)6
Answer: \((x+4)(x-6)\)

27) \(~x^2-8x+15~\)

Two beautiful numbers \(\times\) to make 15 and \(+\) to make \(-\)8
\(-3\times -5=15\) and \(-3+(-5)=-8\)
Our beautiful numbers are \(-\)3 and \(-\)5
Answer: \((x-3)(x-5)\)

28) \(~x^2-5x+6~\)

Two beautiful numbers \(\times\) to make 6 and \(+\) to make \(-\)5
\(-2\times -3=6\) and \(-2+(-3)=-5\)
Our beautiful numbers are \(-\)2 and \(-\)3
Answer: \((x-2)(x-3)\)