Cristian Calude, proba pe echipe, R.IV, P.II
Posted: Tue Nov 18, 2008 4:24 pm
Sa se determine numerele intregi x si y care verifica ecuatia: \( x-y=x^2+y^2 \)
\( x^2-x+y^2+y=0 \)
x - intreg , deci x real
\( \Rightarrow \Delta_x\ge0 \)
\( 1-4y^2-4y\ge0 \Rightarrow 4y^2+4y-1\le0 \Rightarrow (2y+1)^2\le2 \Rightarrow |2y+1|\le \sqrt{2} \Rightarrow -\sqrt{2}\le2y+1\le\sqrt{2} \Rightarrow y \in [\frac{-1-\sqrt{2}}{2};\frac{\sqrt{2}-1}{2}] \cap \mathb{Z} ={-1;0} \)
\( y=-1 \Rightarrow x=0 \) sau \( x=1 \)
\( y=0 \Rightarrow x=0 \) sau \( x=1 \)