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    How do I solve the following equation? I need help in simplifying the roots.




    I need help in simplifying the roots.
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    (Original post by asadmoosvi)
    How do I solve the following equation? I need help in simplifying the roots.




    I need help in simplifying the roots.
    Do you recognize this as a quadratic equation in z?
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    (Original post by SamKeene)
    Do you recognize this as a quadratic equation in z?
    Yes, I know it's a quadratic equation, but I can't simplify the roots for some reason.
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    (Original post by asadmoosvi)
    Yes, I know it's a quadratic equation, but I can't simplify the roots for some reason.
    What did you get for the roots?
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    (Original post by SamKeene)
    What did you get for the roots?
    I get these two roots but how do I simplify them further?

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    (Original post by asadmoosvi)
    I get these two roots but how do I simplify them further?

    So you have \frac{-2i\sqrt{2}\pm i\sqrt{8+8i\sqrt{3}}}{2}

    Factor out \sqrt{8}.

    \frac{-2i\sqrt{2}\pm i\sqrt{8}\sqrt{1+i\sqrt{3}}}{2}

    \frac{-2i\sqrt{2}\pm 2i\sqrt{2}\sqrt{1+i\sqrt{3}}}{2}

    The 2's cancel:

    -i\sqrt{2}\pm i\sqrt{2}\sqrt{1+i\sqrt{3}}

    Factor out..


    -i\sqrt{2}(1\mp\sqrt{1+i\sqrt{3}}  )

    Best I can do ^.^
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    (Original post by SamKeene)
    So you have \frac{-2i\sqrt{2}\pm i\sqrt{8+8i\sqrt{3}}}{2}

    Factor out \sqrt{8}.

    \frac{-2i\sqrt{2}\pm i\sqrt{8}\sqrt{1+i\sqrt{3}}}{2}

    \frac{-2i\sqrt{2}\pm 2i\sqrt{2}\sqrt{1+i\sqrt{3}}}{2}

    The 2's cancel:

    -i\sqrt{2}\pm i\sqrt{2}\sqrt{1+i\sqrt{3}}

    Factor out..


    -i\sqrt{2}(1\mp\sqrt{1+i\sqrt{3}}  )

    Best I can do ^.^
    You can simplify further using polar/exponential form. Which qualification is this for and how much content is there?
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    (Original post by asadmoosvi)
    I get these two roots but how do I simplify them further?

    See my above post. Do you know polar/exponential form of complex numbers?
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    (Original post by morgan8002)
    You can simplify further using polar/exponential form. Which qualification is this for and how much content is there?
    I'm assuming A level, but that's a question to ask OP.

    Out if interest, how would you simplify it further using polar/exp form?
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    (Original post by SamKeene)
    I'm assuming A level, but that's a question to ask OP.

    Out if interest, how would you simplify it further using polar/exp form?
    Sorry, I replied to the most complete working. I have now asked the OP.
    Yes, exp/polar form is covered in FP2.
    Put 1+i\sqrt{3} in (say exp) form: 2e^{\frac{i\pi}{3}}
    Then take the square root: \sqrt{2}e^{\frac{i\pi}{6}}.
    Convert back to cartesian and then tidy up a bit.
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    (Original post by morgan8002)
    Sorry, I replied to the most complete working. I have now asked the OP.
    Yes, exp/polar form is covered in FP2.
    Put 1+i\sqrt{3} in (say exp) form: 2e^{\frac{i\pi}{3}}
    Then take the square root: \sqrt{2}e^{\frac{i\pi}{6}}.
    Convert back to cartesian and then tidy up a bit.
    That's a real neat trick pulling the complex number out the root like that. I'll have to keep it in mind.
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    (Original post by SamKeene)
    That's a real neat trick pulling the complex number out the root like that. I'll have to keep it in mind.
    It is cool, but wouldn't be required for anything easier than FP2.

    Have you done FP2 out of curiosity? Because you seemed to accept my method pretty well.
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    (Original post by morgan8002)
    It is cool, but wouldn't be required for anything easier than FP2.

    Have you done FP2 out of curiosity? Because you seemed to accept my method pretty well.
    Yeah (and FP3/4).

    Speaking of awesome tricks, to return the favor (you might know already), ill share a trick I saw recently.

    You can avoid using an integration by parts on:

    \int \! e^x\cos{3x} \, \mathrm{d}x

    And similar by noting this integrand is the real part of e^x(\cos{3x}+i\sin{3x}) which can be rewritten as e^xe^{3ix} or e^{1+3ix}. Thus:

    \int \! e^{1+3ix} \, \mathrm{d}x = \frac{1}{1+3i}e^{(1+3i)x}+k

    But \frac{1}{1+3i} = \frac{1}{(1+3i)}\frac{(1-3i)}{(1-3i)}=\frac{1-3i}{10}

    Hence:

    \frac{1}{1+3i}e^{(1+3i)x}=\frac{  1}{10}e^x(1-3i)(\cos{3x}+i\sin{3x})

    The required integral is the real part of:

    \int \! e^{(1+3i)x} \, \mathrm{d}x

    -->

    \int \! e^x\cos{3x} \, \mathrm{d}x =\frac{1}{10}e^x (\cos{3x}+3\sin{3x}) + k
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    (Original post by SamKeene)
    Yeah (and FP3/4).

    Speaking of awesome tricks, to return the favor (you might know already), ill share a trick I saw recently.

    You can avoid using an integration by parts on:

    \int \! e^x\cos{3x} \, \mathrm{d}x

    And similar by noting this integrand is the real part of e^x(\cos{3x}+i\sin{3x}) which can be rewritten as e^xe^{3ix} or e^{1+3ix}. Thus:

    \int \! e^{1+3ix} \, \mathrm{d}x = \frac{1}{1+3i}e^{(1+3i)x}+k

    But \frac{1}{1+3i} = \frac{1}{(1+3i)}\frac{(1-3i)}{(1-3i)}=\frac{1-3i}{10}

    Hence:

    \frac{1}{1+3i}e^{(1+3i)x}=\frac{  1}{10}e^x(1-3i)(\cos{3x}+i\sin{3x})

    The required integral is the real part of:

    \int \! e^{(1+3i)x} \, \mathrm{d}x

    -->

    \int \! e^x\cos{3x} \, \mathrm{d}x =\frac{1}{10}e^x (\cos{3x}+3\sin{3x})
    I've done FP2 and FP3 and am taking in May/June. I am doing FP4 currently.


    That is a good trick. Thanks, I hadn't seen that before. Double parts can be tedious.
    You missed a bracket pair on two of the lines, but you seem to have corrected later.
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    (Original post by morgan8002)
    I've done FP2 and FP3 and am taking in May/June. I am doing FP4 currently.


    That is a good trick. Thanks, I hadn't seen that before.
    You missed a bracket pair on two of the lines, but you seem to have corrected later.
    Missed brackets? Where

    FP4 is pretty awesome, if I remember its just Matrices and Vectors which are awesome with the synergies they have with each others and other areas of maths.
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    (Original post by SamKeene)
    Missed brackets? Where

    FP4 is pretty awesome, if I remember its just Matrices and Vectors which are awesome with the synergies they have with each others and other areas of maths.
    It should be (1+3i)x, but in two places you wrote 1+3ix.



    Yeah it's good. I'm still trying to get my head around some areas though.
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    (Original post by morgan8002)
    It should be (1+3i)x, but in two places you wrote 1+3ix.



    Yeah it's good. I'm still trying to get my head around some areas though.
    What kind of stuff have you covered so far/hard to get your head around?

    If you want a book which really covers the FP1-FP4 vectors/matrices stuff well, I suggest Bostock and Chandlers Further Pure Mathematics. It lends itself to a deeper understanding and also explains the relevance of things like orthogonal matrix/diagonalization etc (which most A level textbooks won't).
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    (Original post by asadmoosvi)
    How do I solve the following equation? I need help in simplifying the roots.




    I need help in simplifying the roots.
    the square root requires clever manipulations

    √[8 +8i√3]
    =√[8 +2.4i√3]
    =√[12-4 +2.4i√3]
    =√[(2√3)2 +(2i)2 +2.2i.2√3]
    =√[2i+2√3]2
    =2i+2√3
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    (Original post by SamKeene)
    What kind of stuff have you covered so far/hard to get your head around?

    If you want a book which really covers the FP1-FP4 vectors/matrices stuff well, I suggest Bostock and Chandlers Further Pure Mathematics. It lends itself to a deeper understanding and also explains the relevance of things like orthogonal matrix/diagonalization etc (which most A level textbooks won't).
    I didn't start that long ago. I did a few days before Christmas and then most of last week as well. I've done the first 5 chapters out of 7 of the AQA textbook, but so far I've skipped over the areas I've struggled with, with the intention of going over it later.
    So far it's factorising determinants and vector equations of lines and planes.
    I think the vector equations just needs going over properly once for a few hours, because there are just so many forms etc.. I plan to do that tomorrow after college and get it sorted out properly.
    As for factorising determinants, I don't think I will get it from the textbook, so I'm going to have a look for something online or something.


    Thanks for the suggestion. I will look into it. FP1 has very little matrices and no vectors, it's just FP4 where they put it all.
 
 
 
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