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Math Help - Quadratic Equations #2

  1. #1
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    Quadratic Equations #2

    The roots of the quadratic equation x^2 - √12x + 1 = 0 are c and d.
    Without using calculators, show that 1/c + 1/d = 2 √3 by finding c and d first
    Sum of roots = √12
    Pdt of roots = 1
    I can solve it without finding c and d, but how do I solve using the method specified in the question?
    Last edited by mr fantastic; April 15th 2011 at 01:15 AM. Reason: Title.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    You can easily prove that if r is a root of x^2-\sqrt{12}x+1=0 then, 1/r is also a root.
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  3. #3
    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by Drdj View Post
    The roots of the quadratic equation x^2 - √12x + 1 = 0 are c and d.
    Without using calculators, show that 1/c + 1/d = 2 √3 by finding c and d first
    Sum of roots = √12
    Pdt of roots = 1
    I can solve it without finding c and d, but how do I solve using the method specified in the question?
    i think the questions asks you to use the formula for the roots of the quadratic Quadratic equation - Wikipedia, the free encyclopedia.
    it won't require a calculator.
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  4. #4
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    Quote Originally Posted by Drdj View Post
    The roots of the quadratic equation x^2 - √12x + 1 = 0 are c and d.
    Without using calculators, show that 1/c + 1/d = 2 √3 by finding c and d first
    Sum of roots = √12
    Pdt of roots = 1
    I can solve it without finding c and d, but how do I solve using the method specified in the question?
    Use thequadratic formula to find c and d, of course.
    x= \frac{\sqrt{12}\pm\sqrt{12- 4}}{2} = \frac{2\sqrt{3}\pm\2\sqrt{2}}{2} = \sqrt{3}\pm\sqrt{2}. Take c= \sqrt{3}+ \sqrt{2} and d= \sqrt{3}- \sqrt{2}.
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  5. #5
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    ax^2+bx+c = 0 has the roots as given in post 3.

    ax^2+bx+c = a(x-alpha)(x-beta).

    When x is alpha or beta, the quadratic will be zero.
    The two solutions as per post 3 are alpha and beta (if you want to name them).

    Notice when you multiply out

    (x-alpha)(x-beta)

    you get

    x(x-beta)-alpha(x-beta) = x^2-(beta)x-(alpha)x+(alpha)(beta) = x^2-(alpha+beta)x+(alpha)(beta)

    = x^2-(sum of roots)x+product of roots

    and is simplest when "a" is 1.

    When factoring is easy, it's not always necessary to have to resort to the quadratic formula
    to find the roots as given in post 3.
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