Equation help

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April 5th 2006, 02:31 PM
OReilly

Equation help

Can someone help me to solve this equation for x and y?

$5x^2+5y^2+8xy+2y-2x+2=0$
April 5th 2006, 08:59 PM
earboth

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Quote:

Originally Posted by OReilly

Can someone help me to solve this equation for x and y?

$5x^2+5y^2+8xy+2y-2x+2=0$

Hello,

the bad news first: I cann't show you how to solve this equation. I only can give you the solution, which I found by trial and error: x = 1; y = -1.

Your equation describes how to get the zeros of an elliptical paraboloid. The vertex of this curve is (1, -1, 0). I've attached a diagram to demonstrate my results.

Greetings

EB
April 6th 2006, 12:10 AM
mathemagic

cool
April 6th 2006, 02:03 AM
OReilly

Quote:

Originally Posted by earboth

Hello,

the bad news first: I cann't show you how to solve this equation. I only can give you the solution, which I found by trial and error: x = 1; y = -1.

Your equation describes how to get the zeros of an elliptical paraboloid. The vertex of this curve is (1, -1, 0). I've attached a diagram to demonstrate my results.

Greetings

EB

Diagram looks really nice but I need algebraic solution of this equation. I have been trying to solve it for hours but I can't, it's just not getting to me!
April 6th 2006, 03:05 AM
Rebesques

You cannot get an algebraic solution in the usual sense - every point (x,y) on the parabola is a solution.

Maybe you need to have something else, like y=y(x)? (wherever this is possible).
April 6th 2006, 04:24 AM
topsquark

(Sigh) Once again this morning, I am in a hurry.

As earboth said, there is no single point in general that will be a solution. However, unless I made a mistake (which is quite possible) I am getting that there are NO solutions.

Here's how to algebraically simplify the equation. You can check my Math to see if I was right.

At first glance the equation appears to be describing an ellipse that has been rotated about the origin. However there is a glitch...

First, we have an "xy" term in the equation we need to get rid of. Perform a coordinate rotation:
$x=x' cos\theta - y' sin \theta$
$y=x' sin \theta + y' cos \theta$

When you expand the terms (there are a lot of them!) look for the x'y' term. It will have a coefficient that depends on theta. Choose theta (I had pi/4) such that this coefficient is zero.

I had:
$9x'^2+y'^2+2\sqrt2 y' + 2 = 0$ .

We may complete the square on the y' variable and I got a form that reads:
$9x'^2+(y'^2+\sqrt2 /3)^2 + 28/36 = 0$

This LOOKS like an ellipse, but note that the constant term here is positive. This implies that the axes of the ellispse are imaginary. Hence there is no (real) solution for x' and y' and thus none for x and y.

Again, please check the Math because I did this quickly. At the very least this post gives a way to simplify the original equation.

-Dan
April 6th 2006, 04:51 AM
earboth

Quote:

Originally Posted by topsquark

(Sigh) ...

I had:
$9x'^2+y'^2+2\sqrt2 y' + 2 = 0$ .

We may complete the square on the y' variable and I got a form that reads:
$9x'^2+(y'^2+\sqrt2 /3)^2 + 28/36 = 0$
...
Again, please check the Math because I did this quickly. At the very least this post gives a way to simplify the original equation.
-Dan

Hello,

that's really brillant!

At your last step: You didn't realized, that $y'^2+2\sqrt2 y' + 2$ is a complete square already. Now your equation reads:
$(3x')^2+(y'+\sqrt2)^2 = 0$ .

So you have a sum of two squares which should be zero. It isn't very probable.

Greetings

EB
April 6th 2006, 07:09 AM
ThePerfectHacker

Do this,
$5x^2+5y^2+8xy+2y-2x+2=0$
Express as,
$5y^2+y(8x+2)+(5x^2-2x+2)=0$
Thus,
$y=\frac{-8x\pm \sqrt{-36x^2+72x-36}}{10}$
Thus,
$y=\frac{-8x\pm \sqrt{-36(x-1)^2}}{10}$ .
Which has real solutions where,
$-36(x-1)^2\geq 0$ , thus,
$(x-1)^2\leq 0$ which only happens when,
$x=1$
From here we can solve for $y$ ,
$y=-1$
April 6th 2006, 10:21 AM
topsquark

Ah well. I DID suspect I had screwed up somewhere! :)

-Dan
April 6th 2006, 11:20 AM
DenMac21

This is the best solution in my opinion and most elegant.

$\begin{array}{l} 5x^2 + 5y^2 + 8xy + 2y - 2x + 2 = 0 \\ 4x^2 + 4y^2 + 8xy + x^2 + y^2 + 2y - 2x + 1 + 1 = 0 \\ (2x + 2y)^2 + (x - 1)^2 + (y + 1)^2 = 0 \\ \end{array} $

Now, we have to solve system of equations:
$\begin{array}{l} x + y = 0 \\ x - 1 = 0 \\ y + 1 = 0 \\ \end{array} $

which gives us that solutions are: $x = 1 \wedge y = - 1$
April 6th 2006, 11:32 AM
ThePerfectHacker

Quote:

Originally Posted by DenMac21

This is the best solution in my opinion and most elegant.

$\begin{array}{l} 5x^2 + 5y^2 + 8xy + 2y - 2x + 2 = 0 \\ 4x^2 + 4y^2 + 8xy + x^2 + y^2 + 2y - 2x + 1 + 1 = 0 \\ (2x + 2y)^2 + (x - 1)^2 + (y + 1)^2 = 0 \\ \end{array} $

Now, we have to solve system of equations:
$\begin{array}{l} x + y = 0 \\ x - 1 = 0 \\ y + 1 = 0 \\ \end{array} $

which gives us that solutions are: $x = 1 \wedge y = - 1$

It is elegant and I like it, but I think that my solution with the basic use of the quadradic forumla is simpler. ;)
April 6th 2006, 01:28 PM
topsquark

Well, I think MY solution was the best. Even though it, umm... didn't work. :eek:

-Dan

(It could've, though!)