Question

Fie x,y doua numere rationale pentru care avem $2 \leq x \leq 6$ si $-6 \leq y \leq 2$ . Aratati ca valoarea numarului a = $\sqrt{(3x+2y-22) ^{2} } + |x-y| + \sqrt{(2x+3y+14) ^{2} }$ este patratul unui numar rational care nu depinde de x si y.

Answer 1

$E~o~simpla~problema~de~calcul: \\ \\ y \leq 2~si~2 \leq x \Rightarrow y \leq x \Rightarrow |x-y|=x-y.\\ \\ 3x+2y-22 \leq 3 \cdot 6+2 \cdot 2-22=0 \Rightarrow|3x+2y-22|=-3x-2y+22. \\ \\ 2x+3y+14 \geq 2 \cdot 2+3 \cdot (-6)+14=0 \Rightarrow \\ \\ \Rightarrow |2x+3y+14|=2x+3y+14.$

$a=|3x+2y-22|+|x-y|+|2x+3y+14|= \\ \\ =-3x-2y+22+x-y+2x+3y+14= \\ \\ =36= \\ \\ =6^2.$