I have learned 3 methods of dealing with quadratic expressions.
1: Factorising
2: Quadratic Formula
3: Completing the square
Whilst I can utilise each of these individually, I am not always sure which one to choose. As I understand it, there are situations where some of these methods will work and others will not. How can I recognise which method to use to get the answer I need, without having to use trial and error and end up applying all three if necessary? Is there a means of selection?

Illidan2
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 11102017 11:11

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 11102017 11:18
Look at your equation, if you can find two numbers that will multiply to make the last number and add the middle number (with the single x), then you can factorise.
If not, you can use either of the two methods. Exam questions may ask for a specific method
Completing the square and the Quadratic formula are very similar, but CtS can be used without a calculator, with the QF, you don't have to use a calculator, but if the number underneath the square root can cancel down, then you could use it without a calculator. 
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 11102017 11:23
(Original post by Illidan2)
I have learned 3 methods of dealing with quadratic expressions.
1: Factorising
2: Quadratic Formula
3: Completing the square
Whilst I can utilise each of these individually, I am not always sure which one to choose. As I understand it, there are situations where some of these methods will work and others will not. How can I recognise which method to use to get the answer I need, without having to use trial and error and end up applying all three if necessary? Is there a means of selection?
Personally I avoid completing the square and most students do. Always try factorising first but don't spend too long on it if you really can't see it. Use the quadratic formula if you can't factorise it. If you like you can check that is a square number  if it is then the quadratic factorises. But finding is one of the steps to using the quadratic formula so you could argue that you may as well just use the formula in this case
Also there could be a hint in the question e.g. if the question asks for an answer to 2 d.p. or in surd form then the quadratic formula is the way to go. Are you doing GCSEs?Last edited by Notnek; 11102017 at 11:28. 
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 11102017 11:24
(Original post by Illidan2)
I have learned 3 methods of dealing with quadratic expressions.
1: Factorising
2: Quadratic Formula
3: Completing the square
Whilst I can utilise each of these individually, I am not always sure which one to choose. As I understand it, there are situations where some of these methods will work and others will not. How can I recognise which method to use to get the answer I need, without having to use trial and error and end up applying all three if necessary? Is there a means of selection? 
Illidan2
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 11102017 11:33
Oh okay, I see Suppose I want to obtain the roots of a function f(x). I know for a fact that I can do this via factorisation, but would the quadratic formula/completing the square work in this situation? Each method appears to yield the answer in a different form.
Also why do students avoid completing the square? Is is it suboptimal in the sense that it requires more time to calculate the solution? 
Notnek
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 11102017 11:41
(Original post by Illidan2)
Oh okay, I see Suppose I want to obtain the roots of a function f(x). I know for a fact that I can do this via factorisation, but would the quadratic formula/completing the square work in this situation? Each method appears to yield the answer in a different formAlso why do students avoid completing the square? Is is it suboptimal in the sense that it requires more time to calculate the solution? 
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 11102017 11:45
(Original post by Illidan2)
Oh okay, I see Suppose I want to obtain the roots of a function f(x). I know for a fact that I can do this via factorisation, but would the quadratic formula/completing the square work in this situation? Each method appears to yield the answer in a different form.
Also why do students avoid completing the square? Is is it suboptimal in the sense that it requires more time to calculate the solution? 
HeadmasterCid
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 11102017 11:50
Sorry to be contrary to some of the advice given on here, but I would always try completing the square before using the formula. The main reason why is because the formula is just the general case of completing the square anyway. When you use the formula all you are doing is skipping steps that you would have taken in order to complete the square.
In fact, if you've never seen it before, I recommend looking up the derivation of the quadratic formula  it is just completing the square for ax^2 + bx + c = 0 !!!!
I know it sounds weird to want to take further steps, but it means errors are potentially less likely, and also in the completed square form the equation can tell you some useful information.
ie for (x  p)^2 + q = 0, the vertex (min in this case) is at (p,q).
Being a mathmo though I think I just prefer the fact that is feels far more elegant and pure than just plugging values into a formula.
With that said though I'd always check to see if it can be factorised first. However if I can't see a solution quickly I'll just crack on with completing the square... 
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 11102017 11:56
(Original post by Illidan2)
Oh okay, I see Suppose I want to obtain the roots of a function f(x). I know for a fact that I can do this via factorisation, but would the quadratic formula/completing the square work in this situation? Each method appears to yield the answer in a different form.
Also why do students avoid completing the square? Is is it suboptimal in the sense that it requires more time to calculate the solution?
You should, unless the question specifies otherwise, always give your answer in terms of surds, and so completing the square is preferred to using the formula, although you could always use the formula and simplify until you have the answer (see my previous post  this is exactly the same as completing the square).
All quadratics that have solutions factorise, but not all of them have nice factorisations. Generally, if you can factorise it easily you should, since the formula and CTS will give you the same answers anyhow. If you can't factorise it easily then try CTS or simplifying the formula by hand to get an exact answer with surds. 
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 11102017 11:59
(Original post by HeadmasterCid)
I know it sounds weird to want to take further steps, but it means errors are potentially less likely, and also in the completed square form the equation can tell you some useful information.
By the way I'm definitely not saying students shouldn't use completing the square to solve quadratics. With practice it's just as good as using the formula (I realise that both methods are technically the same ). 
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 11102017 12:11
(Original post by Notnek)
Assuming you're not told which method to use:
Personally I avoid completing the square and most students do. Always try factorising first but don't spend too long on it if you really can't see it. Use the quadratic formula if you can't factorise it. If you like you can check that is a square number  if it is then the quadratic factorises. But finding is one of the steps to using the quadratic formula so you could argue that you may as well just use the formula in this case
Also there could be a hint in the question e.g. if the question asks for an answer to 2 d.p. or in surd form then the quadratic formula is the way to go. Are you doing GCSEs? 
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 11102017 12:13
(Original post by Notnek)
I'm not sure why you think errors would be less likely when using completing the square? I would think the opposite might be true for GCSE students.
By the way I'm definitely not saying students shouldn't use completing the square to solve quadratics. With practice it's just as good as using the formula (I realise that both methods are technically the same ).
For me, plugging numbers into a formula makes errors likely. Whereas actually going through the steps (as you do with CTS) will make errors much less likely. But I understand that for others (particularly going back down to GCSE or below) adding in further steps gives a greater chance of error.
It obviously depends on the person, but for those seeking mathematical enlightenment I would recommend CTS over the formula any day! (Meme not my own) 
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 11102017 12:18
I didn't know the quadratic formula was part of the GCSE spec. I don't remember it, and this is all new to me, you see. I remember factorisation, but not CTS or the Quadratic Formula. I thought they were introduced at ALevel. This is somewhat discouraging :P I do hope I'll be able to cope as the material progresses.

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 11102017 12:18
(Original post by Illidan2)
I am selfteaching myself the new Edexcel alevel maths spec(2017). It has been over five years since I studied GCSE maths. I do not have a teacher, only myself, a textbook, and TSR. The textbook implies that for a function f(x), to draw a quadratic graph I need the roots of the function for the x coordinates, and with (p, q) as my turning point. The textbook tells me to complete the square to get the turning point. When I was referring to the answer in different forms I was referring to things such as precisely this, as I only find that my answer is in this form when I complete the square, and that the roots of the function only seem apparent to me when I factorise etc.
Also you could factorise or use the formula and find the x value of the T.P. by averaging the x values of the roots. Then plug this in to find y. Not very elegant thoughLast edited by HeadmasterCid; 11102017 at 12:31. 
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 11102017 12:18
(Original post by Illidan2)
I am selfteaching myself the new Edexcel alevel maths spec(2017). It has been over five years since I studied GCSE maths. I do not have a teacher, only myself, a textbook, and TSR. The textbook implies that for a function f(x), to draw a quadratic graph I need the roots of the function for the x coordinates, and with (p, q) as my turning point. The textbook tells me to complete the square to get the turning point. When I was referring to the answer in different forms I was referring to things such as precisely this, as I only find that my answer is in this form when I complete the square, and that the roots of the function only seem apparent to me when I factorise etc.
By the way, please continue to post questions on TSR whenever you are stuck  selfteaching can be hard without a teacher. And I recommend watching ExamSolutions videos if you're not already. 
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 11102017 12:30
(Original post by Illidan2)
I didn't know the quadratic formula was part of the GCSE spec. I don't remember it, and this is all new to me, you see. I remember factorisation, but not CTS or the Quadratic Formula. I thought they were introduced at ALevel. This is somewhat discouraging :P I do hope I'll be able to cope as the material progresses. 
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 11102017 12:38
(Original post by HeadmasterCid)
Some GCSE syllabi include CTS, I think. I'm fairly sure all include the quadratic formula. That way there is always a failsafe if they can't factorise. 
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 11102017 12:45
(Original post by Notnek)
The content is the same across all exam boards and they all include CTS and have done since I can remember. And nowadays they include harder questions where the leading coefficient is greater than 1  this was only part of A Level a few years ago.
Well, I think it is good that CTS is being rolled out at GCSE, down with the quadratic formula!
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