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Thread: How to extract square root of three or more terms?

  1. #1
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    How to extract square root of three or more terms?

    Qu: Dear Math helpers, I have been tackling a book by Durell called School Algebra to improve my math skills and to improve my algebraic manipulation skills.
    I have learn how to square trinomial such as (a+b+c)(a+b+c) etc. However, I have hit a bump.

    I have shared the link of the book below and a picture of page. The place where i am stuck is page 283. The problem is that I don't understand the method and the example given by author.
    The example was: Extact the square root of $x^4 - 6x^3 + 19x^2 - 30x + 25$


    The book can be downloaded on the archive site (oso no copyrights issues) : https://archive.org/details/durellsschoolal01duregoog

    How to extract square root of three or more terms?-problem.jpg

    Thank you ver much for your help
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  2. #2
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    Re: How to extract square root of three or more terms?

    I'm not familiar with the author's process ... I'd have to go over it if I have time.

    I would approach the problem in this manner ...

    If it is reducible, the quartic polynomial will have a quadratic square root, i.e. ...

    $x^4-6x^3+19x^2-30x+25 = (ax^2+bx+c)^2$

    $x^4-6x^3+19x^2-30x+25 = a^2x^2 + 2abx^3 + (2ac+b^2)x^2 + 2bcx + c^2$

    matching coefficients, it's easy to see $a=1$ and $c=5$

    $2b = -6 \implies b = -3$

    so ...

    $(x^2-3x+5)^2 = x^4-6x^3+19x^2-30x+25$

    now, would $(-x^2 + 3x -5)^2$ also work?
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  3. #3
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    Re: How to extract square root of three or more terms?

    Quote Originally Posted by skeeter View Post
    I'm not familiar with the author's process ... I'd have to go over it if I have time.

    I would approach the problem in this manner ...

    If it is reducible, the quartic polynomial will have a quadratic square root, i.e. ...

    $x^4-6x^3+19x^2-30x+25 = (ax^2+bx+c)^2$

    $x^4-6x^3+19x^2-30x+25 = a^2x^2 + 2abx^3 + (2ac+b^2)x^2 + 2bcx + c^2$

    matching coefficients, it's easy to see $a=1$ and $c=5$

    $2b = -6 \implies b = -3$

    so ...

    $(x^2-3x+5)^2 = x^4-6x^3+19x^2-30x+25$

    now, would $(-x^2 + 3x -5)^2$ also work?
    thank you very much
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