Sum of 2 Squares

• September 26th 2008, 09:03 AM
ViperRobK
Sum of 2 Squares
How do you express x^2 + xy + y^2 as the sum of two squares

Any help much appreciated.

ViperRobK
• September 26th 2008, 11:03 AM
Soroban
Hello, ViperRobK!

Quote:

How do you express $x^2 + xy + y^2$ as the sum of two squares?
There is a way . . . but it's not "pleasant".

Subtract and add $3xy\!:\quad x^2 + xy \:{\color{red}-\;3xy} + y^2 \;{\color{red}+\; 3xy}$

. . . . . .and we have: . $x^2 - 2xy + y^2 + 3xy$

. . . . . . . and finally: . $(x-y)^2 + (\sqrt{3xy})^2$