Question: Quadratische Ergänzung

[Gutwiler Hans]

Hallo,

can anyone help me for this "quadratischen Ergänzung":

x^2 + 2x = y^2 + 2y

(x+1)^2 + 1 = (y+1)^2 + 1 => is it correct?

Now, how we comes on this result?

y = -x-2


Thanks a lot for your help

Hans

[GerdW]

Hi Hans,

your two formulas can be transformed as:
1) x^2 + 2x = y^2 + 2y => x(x+2) = y(y+2)

2) (x+1)^2 + 1 = (y+1)^2 + 1 => x(x+2)+2 = y(y+2)+2

So essentially both formulas are the same (just an offset of 2) and other possible result should be x=y...

When you insert y=-x-2 into 1st formula you get y(y+2) = (-x-2)(-x-2+2) = (-x-2)*-x = x^2+2x = x(x+2). This will be the second solution for your 1st formula...

[GerdW]

Hi Hans,

just refreshed "quadr. Ergänzung" on wikipedia:

a) x^2 +2x = y^2 + 2y

b) x^2 + 2x +1 -1 = y^2 + 2y +1 -1

c) (x+1)^2 -1 = (y+1)^2 -1

That's how it should be done according to wikipedia...


Message Edited by GerdW on 05-23-2008 11:41 AM