How do you prove that x^2+y^2-2xy+x-y+1 is greater than 0?

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- Mar 10th 2013, 01:11 AMRuyHayabusaProving
How do you prove that x^2+y^2-2xy+x-y+1 is greater than 0?

- Mar 10th 2013, 01:26 AMProve ItRe: Proving
You can't.

$\displaystyle \displaystyle \begin{align*} x^2 - 2\,x\,y + y^2 + x - y + 1 &= \left( x - y \right)^2 + x - y + 1 \end{align*}$

All that you can say for sure is that $\displaystyle \displaystyle \begin{align*} \left( x - y \right)^2 \end{align*}$ is nonnegative. The entire quantity will only be positive if $\displaystyle \displaystyle \begin{align*} x - y + 1 > 0 \end{align*} $, which you are not guaranteed. - Mar 10th 2013, 06:25 AMNehushtanRe: Proving
- Mar 10th 2013, 09:41 AMMINOANMANRe: Proving
Excellent proof ...My congratulations to Nehushtan

Minoas