• May 15th 2010, 10:43 PM
demode
http://img695.imageshack.us/img695/843/73734670.gif

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

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

$\displaystyle = 4x_1^2 + 5x^2_2 +6x_1x_2$

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...)
• May 16th 2010, 03:30 AM
HallsofIvy
Yes, what you have so far, $\displaystyle 4x_1^2+ 6x_1x_2+ 5x_2^2$, is correct.

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

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