# Quadratics : which method to use?Watch

Thread starter 1 year ago
#1
I have learned 3 methods of dealing with quadratic expressions.

1: Factorising
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?
0
1 year ago
#2
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.
0
1 year ago
#3
(Original post by Illidan2)
I have learned 3 methods of dealing with quadratic expressions.

1: Factorising
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?
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?
0
1 year ago
#4
(Original post by Illidan2)
I have learned 3 methods of dealing with quadratic expressions.

1: Factorising
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?
You use the quadratic formula if the question asks you to give the answer to any decimal places (2,3 etc).
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Thread starter 1 year ago
#5
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?
0
1 year ago
#6
(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
Do you have an example? If a quadratic factorises then I recommend factorising. Using the formula will also work and I'm not sure why you think the answers are 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?
If a student uses the method regularly then completing the square will probably be just as quick as using the formula. But the formula is easier to use for most students so they don't have much practice. Completing the square is normally easier when the coefficient is even.
0
1 year ago
#7
(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?
Technically speaking all methods will always work but I think most of the time it's just quicker and easier to use the formula - unless it factorises nicely.
0
1 year ago
#8
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...
0
1 year ago
#9
(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?
All three should give you answers in the same format. However if your calculator cannot output surds then it will just give you a horrible decimal answer when you plug values in (unless it factorises!).

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.
0
1 year ago
#10
(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.
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 ).
0
Thread starter 1 year ago
#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?
I am self-teaching myself the new Edexcel a-level 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.
0
1 year ago
#12
(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 ).
That's why I said 'potentially' 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) 0
Thread starter 1 year ago
#13
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 A-Level. This is somewhat discouraging :P I do hope I'll be able to cope as the material progresses.
0
1 year ago
#14
(Original post by Illidan2)
I am self-teaching myself the new Edexcel a-level 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.
There are certainly other ways to find the turning point, such as differentiation. However I think differentiation is seen as more effort (arguably it is quicker and easier), and slightly more 'advanced'. CTS is a really nice way to see where the vertex is, but does require a little bit of understanding as to why should give you a T.P. of (-p,q).

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 though 0
1 year ago
#15
(Original post by Illidan2)
I am self-teaching myself the new Edexcel a-level 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.
Sorry I read your post twice and I am not sure what you mean (maybe someone else understands). You are talking about finding roots and finding turning points but I'm not sure how these relate to each other in your post. I think if you post specific examples it will be easier to understand what you mean.

By the way, please continue to post questions on TSR whenever you are stuck - self-teaching can be hard without a teacher. And I recommend watching ExamSolutions videos if you're not already.
0
1 year ago
#16
(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 A-Level. This is somewhat discouraging :P I do hope I'll be able to cope as the material progresses.
Some GCSE syllabi include CTS, I think. I'm fairly sure all include the quadratic formula. That way there is always a fail-safe if they can't factorise.
0
1 year ago
#17
(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 fail-safe if they can't factorise.
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.
1
1 year ago
#18
(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.
Fair. I did an AQA GCSE Maths over ten years ago and couldn't remember CTS being involved, but I'm probably just getting old and losing my memory!

Well, I think it is good that CTS is being rolled out at GCSE, down with the quadratic formula!
0
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