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Math Help - Quadratic Forms

  1. #1
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    Quadratic Forms



    What does it mean to write Q as a function of x_1 and x_2? Is the following what they are asking for:

    Q(x) = (x_1 x_2) \begin{pmatrix}4 & 3 \\ 3 & 5 \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \end{pmatrix}

    = 4x_1^2 + 5x^2_2 +6x_1x_2

    Is this correct so far? And how do I need to "complete the square" to show that Q(x)>0? (I don't have any notes on this...)
    Last edited by demode; May 16th 2010 at 12:03 AM.
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  2. #2
    MHF Contributor

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    Yes, what you have so far, 4x_1^2+ 6x_1x_2+ 5x_2^2, is correct.

    And you "complete the square" just the way you learned many years ago in elementary algebra: To make 4x_1^2+ 6x_1x_2= 4(x_1^2+ (3/2)x_1x_2) a "perfect square", you have to add and subtract \left(\frac{\frac{3}{2}x_2}{2}\right)^2= \frac{9}{16}x_2^2.

    4x_1^2+ 6x_1x_2+ \frac{9}{16}x_2^2- \frac{9}{16}x_2^2+ 5x_2^2= 4(x_1+ \frac{3}{4}x_2)^2+ \frac{71}{16}x_2^2 which, as a sum of squares, is never negative.
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