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Math Help - Completing the square.

  1. #1
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    Completing the square.

    A level Mathematics. Use the method of completing the square to solve these quadratic equations. Give your answer in the form a(+or-)b root "n" where a and b are rational, and n is an integer.
    a. x^2+4x-1=0
    d. x^2-8x-3=0

    Please help, as I do not understand fully how to do this.
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  2. #2
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    Quote Originally Posted by greatersanta616 View Post
    A level Mathematics. Use the method of completing the square to solve these quadratic equations. Give your answer in the form a(+or-)b root "n" where a and b are rational, and n is an integer.
    a. x^2+4x-1=0
    d. x^2-8x-3=0

    Please help, as I do not understand fully how to do this.
    (a)

    (x+p)^2=(x+p)(x+p)=x(x+p)+p(x+p)=x^2+2xp+p^2

    Therefore, in x^2+4x=1 the "2p" term is 4, so p is 2.

    But... (x+2)^2=x^2+4x+4

    Hence

    x^2+4x=1\Rightarrow\ x^2+4x+4=1+4

    (x+2)^2=5

    x+2=\pm\sqrt{5}

    x=-2\pm\sqrt{5}

    Try (d)
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  3. #3
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    (d) x^2-8x-3=0
    So..
    (x-4)^2-19=0
    Therefore.
    (x-4)^2=19
    henceforth...
    x-4=(+or-)root 19
    and...
    x=4(+or-)root 19

    Is this ok?
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  4. #4
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    Quote Originally Posted by greatersanta616 View Post
    (d) x^2-8x-3=0
    So..
    (x-4)^2-19=0
    Therefore.
    (x-4)^2=19
    henceforth...
    x-4=(+or-)root 19
    and...
    x=4(+or-)root 19

    Is this ok?
    Yes, that's the completing the square technique..you got it fast!

    I think that "a" and "b" being rational, would apply more to the case of using the quadratic formula,
    however, those values, while being integers, are rational anyway.
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  5. #5
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    Quote Originally Posted by Archie Meade View Post
    Yes, that's the completing the square technique..you got it fast!

    I think that "a" and "b" being rational, would apply more to the case of using the quadratic formula,
    however, those values, while being integers, are rational anyway.
    Thank you very much for this. It really helped.
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