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Math Help - The nature of the roots of a quadratic equation

  1. #1
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    The nature of the roots of a quadratic equation

    Hi,
    May I please have your help in solving the problem below?
    Thanks.

    Show that the roots of the equation x^2 - (a + d)x + (ad - b^2) = 0 are real.
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  2. #2
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    Re: The nature of the roots of a quadratic equation

    Have you tried checking the discriminant of this quadratic equation?
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    Re: The nature of the roots of a quadratic equation

    Yes Sir, I have. b^2 - 4ac greater than or equal to zero;

    a = 1, b = a+d, c = ad - b^2;

    b^2 = (a + d)^2 = a^2 + 2ad + d^2;

    - 4ac = - 4(1)(ad - b^2) = - 4ad + 4b^2;

    However, how does one prove that a^2 - 2ad + d^2 + 4b^2 is greater than or equal to zero?
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    Re: The nature of the roots of a quadratic equation

    Try factoring the first 3 terms as the square of a binomial...then you will have the sum of two squares...
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    Re: The nature of the roots of a quadratic equation

    Quote Originally Posted by Seaniboy View Post
    Show that the roots of the equation x^2 - (a + d)x + (ad - b^2) = 0 are real.
    You are making it too hard.
    \\(a+d)^2-4(1)(ad-b^2)\\=a^2+2ad+d^2-4ad+4b^2\\=a^2-2ad+d^2+4b^2\\=(a-d)^2+4b^2
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    Re: The nature of the roots of a quadratic equation

    Thanks Plato.
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    Re: The nature of the roots of a quadratic equation

    Thanks Mark.
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