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I can't find the difference of two squares. I don't understand :( watch

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    (Original post by steve2005)
    You were the one that said that completing the square was not necessary or useful and not worth learning. I point out that it is covered in GCSE.



    You can do that on sight just to say. Just multiply the co-effs of the x and y by -0.5. The centre is (-2,3)


    That said, I do agree completing the square certainly does have its uses, and refusing to learn it is just stupid, considering it's not that difficult. There may be other ways of finding the minima/maxima (differentiation is the one that comes to mind here), but below is an example of where you need completing the square to answer. This comes directly from an undergraduate level engineering textbook my Physics teacher showed me...

    (Original post by didgeridoo12uk)
    studying engineering, not maths.

    and you've still failed to show me the point of it.
    .. so whoever said it is never used in engineering at degree level would be wrong.

    To integrate a function of the form \frac{1}{ax^2+bx+c}, you need to first express the bottom line of the fraction as a difference of two squares, i.e. in the form \frac{1}{M^2-N^2}. The best way (possibly the ONLY way) of doing this is through completing the square.

    For example\frac{1}{x^2+2x-4}
    Now complete the square to get \frac{1}{(x+1)^2-5}
    =\frac{1}{(x+1)^2-(\sqrt5)^2}

    Which means that M=x+1 and N=\sqrt5

    Which you can now integrate!

    \int \frac{1}{M^2-N^2}\ dx
    = \frac{1}{2N}\ln\frac{M+N}{M-N}+C
    =\frac{1}{2\sqrt5}\ln\frac{x+1+\  sqrt5}{x+1-\sqrt5}+C


    So the idea that completing the square is useless is in fact wrong. Even if you'd rather use another method to find the minima/maxima of parabolas, the best way of integrating a function that is a 1/quadriatic involves completing the square. :cool:
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    (Original post by innerhollow)


    You can do that on sight just to say. Just multiply the co-effs of the x and y by -0.5. The centre is (-2,3)


    That said, I do agree completing the square certainly does have its uses, and refusing to learn it is just stupid, considering it's not that difficult. There may be other ways of finding the minima/maxima (differentiation is the one that comes to mind here), but below is an example of where you need completing the square to answer. This comes directly from an undergraduate level engineering textbook my Physics teacher showed me...



    .. so whoever said it is never used in engineering at degree level would be wrong.

    To integrate a function of the form \frac{1}{ax^2+bx+c}, you need to first express the bottom line of the fraction as a difference of two squares, i.e. in the form \frac{1}{M^2-N^2}. The best way (possibly the ONLY way) of doing this is through completing the square.

    For example\frac{1}{x^2+2x-4}
    Now complete the square to get \frac{1}{(x+1)^2-5}
    =\frac{1}{(x+1)^2-(\sqrt5)^2}

    Which means that M=x+1 and N=\sqrt5

    Which you can now integrate!

    \int \frac{1}{M^2-N^2}\ dx
    = \frac{1}{2N}\ln\frac{M+N}{M-N}+C
    =\frac{1}{2\sqrt5}\ln\frac{x+1+\  sqrt5}{x+1-\sqrt5}+C


    So the idea that completing the square is useless is in fact wrong. Even if you'd rather use another method to find the minima/maxima of parabolas, the best way of integrating a function that is a 1/quadriatic involves completing the square. :cool:
    Partial Fractions?
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    (Original post by didgeridoo12uk)
    studying engineering, not maths.

    and you've still failed to show me the point of it.


    it's like the people the memorise every single formula that could be of any use. rather than just learning the main ones, and how to derive the rest from first principles
    Useful for some integrals?
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    (Original post by didgeridoo12uk)
    it's like the people the memorise every single formula that could be of any use. rather than just learning the main ones, and how to derive the rest from first principles
    Like how you can derive the quadratic formula by completing the square?

    It's useful to have a lot of the results memorised rather than having to work them out: not only does it save time, but it's also often possible to make a question much easier by spotting something in it that's similar to something you know. I'm not advising that you memorise everything you're ever taught, but it makes sense to know some of it.
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    (Original post by harr)
    Like how you can derive the quadratic formula by completing the square?

    It's useful to have a lot of the results memorised rather than having to work them out: not only does it save time, but it's also often possible to make a question much easier by spotting something in it that's similar to something you know. I'm not advising that you memorise everything you're ever taught, but it makes sense to know some of it.
    i never said you can derive the quadratic formula by completing the square ? wtf? you can however derive the quadratic formula quite easily if you want to

    and its a much more useful skill to prove stuff from first principles. alot of the time the equations you'll know are only true for specific circumstances, so you could end up getting the wrong answers.

    and outside a'level questions aren't the same year on year, theres a huge amount of variation, and you need to understand the basics, and have a good knowledge of whats actually going on. thats far more important that learning tonnes of equations
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    (Original post by didgeridoo12uk)
    i never said you can derive the quadratic formula by completing the square ? wtf? you can however derive the quadratic formula quite easily if you want to

    and its a much more useful skill to prove stuff from first principles. alot of the time the equations you'll know are only true for specific circumstances, so you could end up getting the wrong answers.

    and outside a'level questions aren't the same year on year, theres a huge amount of variation, and you need to understand the basics, and have a good knowledge of whats actually going on. thats far more important that learning tonnes of equations
    I think you are just trying to wind people up OR you really don't know anything useful( in a mathematical sense)

    Are you studying for a degree at Cambridge?
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    (Original post by steve2005)
    I think you are just trying to wind people up OR you really don't know anything useful( in a mathematical sense)

    Are you studying for a degree at Cambridge?
    yes i am

    and i know plenty of useful stuff in a mathematical sense, otherwise i'd be unable to do my work here


    and i personally don't think you know anything particularly useful in a mathematical sense
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    (Original post by didgeridoo12uk)
    i never said you can derive the quadratic formula by completing the square ? wtf? you can however derive the quadratic formula quite easily if you want to

    and its a much more useful skill to prove stuff from first principles. alot of the time the equations you'll know are only true for specific circumstances, so you could end up getting the wrong answers.
    Just curious, how would you derive the quadratic formula without completing the square?
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    (Original post by didgeridoo12uk)
    i never said you can derive the quadratic formula by completing the square ? wtf?
    Correct -- I said it.

    My point was that the quadratic equation is just a consequence of completing the square. If you're memorising a few things so that you can derive others it makes more sense to learn how to complete the square.
    and its a much more useful skill to prove stuff from first principles. alot of the time the equations you'll know are only true for specific circumstances, so you could end up getting the wrong answers.
    That's why you know how to derive them and only use them when they'll work.
    and outside a'level questions aren't the same year on year, theres a huge amount of variation, and you need to understand the basics, and have a good knowledge of whats actually going on. thats far more important that learning tonnes of equations
    But that doesn't mean that they aren't both important. I agree that many people take it too far though.
    (Original post by Narev)
    Just curious, how would you derive the quadratic formula without completing the square?
    Wikipedia gives this option, though it's a bit more complicated.
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    (Original post by Narev)
    Just curious, how would you derive the quadratic formula without completing the square?
    using lagrange resolvents, although its a bit beyond a'level

    but can also be applied to cubic equations, and i think quartic, although never done the proofs for them myself
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    Fair enough. That's an alternative way to derive the quadratic formula (which is admittedly much more complicated than completing the square), abd Lagrangian resolvents can be used to derive other formulas as well.

    That doesn't necessarily mean that completing the square is useless though. There are a variety of ways to solve problems / deriving proofs / expressions from first principles, and at times, completing the square is an easier method, considering you need to know virtually nothing about how to complete the square (of course, drawing a diagram of a rectangle / square) helps if you've forgotten how to do that. But I digress.

    I suppose I might as well add on that knowing how to complete the square is one of the more useful ways - perhaps faster too in say, classifying certain types of quadric surfaces by rewriting the equation in a certain form, and also showing certain quadratic forms are non-negative definite or positive definite. Of course you could always put the equation in matrix form, but that's more of memorizing formula and not proving from uh..first principles...

    And completing the square helps when looking at the pdf / MGFs of a Gaussian distribution too.

    *I do believe in memorizing very few things. But I also like to work with 'tools' that allow me to get to the solution the quickest and most efficient way possible.
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    (Original post by didgeridoo12uk)
    using lagrange resolvents, although its a bit beyond a'level

    but can also be applied to cubic equations, and i think quartic, although never done the proofs for them myself
    Just out of curiosity, did you seriously not bother learning how to complete the square for C1?
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    (Original post by didgeridoo12uk)
    using lagrange resolvents, although its a bit beyond a'level

    but can also be applied to cubic equations, and i think quartic, although never done the proofs for them myself
    lol come on now... if you just remember this small thing (the procedure for completing the square), you can save yourself a whole load of time and trouble. i personally think you are trying to show everyone how clever you are by not following the standard way in which certain problems are solved. you say you are studying engineering; well, then you of all people should be looking for the quickest way to solve a problem. after all, if you write a program that does a task with half the number of lines as before, you've effectively doubled your performance.
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    (Original post by idunnowhattoputhere)
    lol come on now... if you just remember this small thing (the procedure for completing the square), you can save yourself a whole load of time and trouble. i personally think you are trying to show everyone how clever you are by not following the standard way in which certain problems are solved.
    I must take issue with you IF you think he is clever. I invite you to count the number of errors in his posts. He thinks alot is one word. I have never seen A-level written as a'level.

    And the idea that he never bothered to learn ' completing the square' is ridiculous. As a 'genius' he may now prefer to use other methods BUT what method did he use when he was a boy.
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    It's understandable though :P You must understand that completing the square is relatively difficult to understand conceptually at GCSEs, and one tends to skip that to use 'plug and play' methods like quadratic formulae, simple differentiation to find gradients, etc. So it is possible to never bother to learn 'completing the square'. But nevertheless, 'completing the square' is an essential tool in mathematics, just maybe not the most 'direct' and 'efficient' until perhaps A levels / Uni.
 
 
 
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