Which method do you use to factorise hard quadratics?

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Poll: Which method do you use?
Inspection / trial and error (6)
40%
The "common" AC method (0)
0%
The "alternative" AC method (0)
0%
The perfect square method (1)
6.67%
The calculator/cheat method (5)
33.33%
Something else (specify in the comments) (3)
20%
Prince Philip
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Report Thread starter 11 months ago
#1
As a tutor I have come across lots of weird and wonderful methods used by students to factorise quadratics where the 'a' coefficient is > 1 e.g.

6x^2 + 13x + 6

Which method do you use? Take part in the poll or post below if your method isn't listed - I'm always interested to hear about new methods and it's nice to have all the different methods in one place so students can choose their favourite

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Some brief notes on the methods I've given so you can identify them.

Methods

Inspection/trial and error : Try factors of the first and last terms inside the brackets until you end up with the correct answer. This method requires lots of practice.

The "common" AC method : Start by multiplying the 'a' and 'c' so in the case above it would give 36 then look for two numbers that add to give 13 and multiply to give 36. They are 4 and 9 so split the middle term to give 6x^2+4x + 9x + 6. Then factorise by grouping to give 2x(3x+2) + 3(3x+2) = (2x+3)(3x+2).

The "alternative" AC method : Start the same as above but then put the numbers you found straight into the brackets with 'ax' at the front of both e.g. (6x+9)(6x+4). Then "cancel"/divide one or both of the brackets, here you do both to give (2x+3)(3x+2).

The perfect square method : Start by multiplying/dividing by 'a' so e.g. 6x^2 + 13x + 6 changes into \frac{1}{6}\left[(6x)^2 + 13(6x) + 36\right] and then factorise the easier quadratic y^2+13y+6.

The calculator/cheat method : Use your calculator to solve the equation 6x^2 + 13x + 6 = 0 which gives x=-\frac{3}{2} and x=-\frac{2}{3}. Then use your knowledge of quadratic equations to work backwards and end up with (2x+3)(3x+2). Of course this can't be used in GCSE Paper 1!
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ghostwalker
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#2
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#2
The quadratic formula.

Or is that not GCSE?
Last edited by ghostwalker; 11 months ago
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laurawatt
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I find inspection works best for me, but the a-levels calculator has given me the luxury of it doing everything for me :lol:

We only ever got taught the inspection/trial and error method in school so I find the AC methods really interesting!
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PepeTheFroggi
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depends on the question. if it says to give to significant figures then i do quadratic formula else i do complete the square or something
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Prince Philip
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#5
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(Original post by PepeTheFroggi)
depends on the question. if it says to give to significant figures then i do quadratic formula else i do complete the square or something
I mean if a question just says "factorise this quadratic" so no solving equations.
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PepeTheFroggi
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(Original post by Sir Cumference)
I mean if a question just says "factorise this quadratic" so no solving equations.
aah whoops didnt read. then id do complete the square
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Obolinda
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Don't think mine is listed
First I have to choose the 2 number that make 4x^2 !!!
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Muttley79
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(Original post by Sir Cumference)
I mean if a question just says "factorise this quadratic" so no solving equations.
It won't though and even if it did you can always use the formula and convert your answers. (x - a)(x - b) and multiply up.
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Prince Philip
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(Original post by Obolinda)
Don't think mine is listed
First I have to choose the 2 number that make 4x^2 !!!
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That's the inspection/trial and error method I mentioned
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Obolinda
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(Original post by Sir Cumference)
That's the inspection/trial and error method I mentioned
Oh

(Original post by ghostwalker)
The quadratic formula.

Or is that not GCSE?
We do use the quadratic formula (that's the calculator method) but paper 1 is non calculator so we need to be able to factorise(not just solve) without.
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Obolinda
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(Original post by laurawatt)
I find inspection works best for me, but the a-levels calculator has given me the luxury of it doing everything for me :lol:

We only ever got taught the inspection/trial and error method in school so I find the AC methods really interesting!
I got the inspection method from my teacher and the AC method from Mathswatch - it is very long!
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Prince Philip
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(Original post by Muttley79)
It won't though and even if it did you can always use the formula and convert your answers. (x - a)(x - b) and multiply up.
I don't see why a GCSE exam question won't ask a student to factorise a quadratic. Yes that's another method that can be used.
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Muttley79
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(Original post by Sir Cumference)
I don't see why a GCSE exam question won't ask a student to factorise a quadratic. Yes that's another method that can be used.
Not one with coefficient like that.
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Prince Philip
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(Original post by Muttley79)
Not one with coefficient like that.
E.g 2018 OCR Paper 4 question 16:

https://www.ocr.org.uk/Images/528854...er-paper-4.pdf
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username4952620
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#15
The X method
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ghostwalker
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#16
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(Original post by Obolinda)
We do use the quadratic formula (that's the calculator method) but paper 1 is non calculator so we need to be able to factorise(not just solve) without.
You can still use the formula without a calculator. The sort of questions that would occur on a non-calculator isn't going to produce surds, you just need proficiency at arithemetic and reasonable knowledge at recognizing perfect squares.
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Obolinda
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(Original post by ghostwalker)
You can still use the formula without a calculator. The sort of questions that would occur on a non-calculator isn't going to produce surds, you just need proficiency at arithemetic and reasonable knowledge at recognizing perfect squares.
I guess, yes. :dontknow: I find the quadratic formula way of factorising hard quadratics confusing personally tho

Oh and some non-calculator questions may produce surds but the question asks you to solve by completing the square
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Dancer2001
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#18
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I used to use the ‘common’ AC method at gcse, but found it too time consuming when I got to A level. Then I started using the calculator/cheat method, and using the quadratic formula if I didn’t have a calculator (which probably takes even longer).
I can usually do it by inspection now, which I used to find too difficult. I learnt to do it when factorising cubics and quartics.
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Muttley79
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(Original post by Sir Cumference)
E.g 2018 OCR Paper 4 question 16:

https://www.ocr.org.uk/Images/528854...er-paper-4.pdf
3 is prime which makes a huge difference - your example did not have that; hence my comment
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Prince Philip
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(Original post by Muttley79)
3 is prime which makes a huge difference - your example did not have that; hence my comment
Ok change the example in my post to 3x^2+11x-20 and the thread and subsequent posts still apply. But it's important to note that the DFE content says that students need to "factorise quadratics of the form ax^2+bx+c" and doesn't say anything about a being prime only. It would be very risky for students to assume that nothing as hard as the question in my OP will come up in the Higher papers.
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