I have a discrete math assignment which says:
Is the following statement true or false? Explain.
For any real numbers x, y, the number 9x2 + 4y2 + 6xy − 3x − 2y + 1 is not negative.
I am pretty sure that this is true (I wrote a program that went through a lot of the possibilities and it was never negative). However, I do not have a vigorous proof that this is true. If anyone can help me get started on this proof it would be great. I talked to my prof and he said the best way was to not break it into cases, and that the other way is really easy. I apparently just can't see it. Thanks in advance for anyones help into this. I do not necessarily want the proof done for me, but even an idea where to start would be great.
I am still confused about yours.... my weakest point in math is completing the square (I know I know... its the easiest thing to do... but its the one thing I can not do). However, I think you might have made a mistake in your square completion on the first line. I am still trying to comprehend it.. but the "dreaded" square completion is taking me time to try to figure out :P
Edit: I guess I mean between the first and second line.
Edit: There really is no error... like I said Completing Square + me = weak
The technique is to isolate the terms containing x. Then completing the square.
You'll be left only with terms containing y and constants... So complete again the square.
It's called "Gauss reduction", litteral translation from French... I don't know the name in English
Well, no one is dumb. All this working comes with habits(PS. Also, thanks for saying I am not dumb lol)
And what about gentlewoman ?I really do thank you both for your help gentlemen.
Ah, thanks. That makes sense. I think it is actually called Gauss reduction in English too (I have heard of it... but was not 100% sure what it was). I will have to try that to see what comes of it. When I started doing it I just started from the left and was like hmm... this doesn't work.
Also, thanks gentlewoman as well. I am really sorry... I usually do use both... I am ashamed
Consider , where k > 0.
To find the curve (if any) that this equation defines (that is, set of points (if any) that satisfy the equation):
a = 9, b = 3, c = 4 so the minor discriminant therefore the curve is either an ellipse, a single point or there's no curve.
No curve if (and I can't be bothered doing the latex) the product of the determinant of the matrix
9, 3, -1.5
3, 4, -1
-1.5, -1, 1 + k
(that is, the major discriminant) and a + c is greater than zero:
(13) (27(k + 2/3)) is greater than zero when k > -2/3. Since by definition k > 0, the product is always greater than zero.
Therefore cannot be less than zero.