Question

$Rezolvati~in~Z~ecuatia:~x^2+y^2-5x-3y+8=0$

Answer 1

$Inmultim~ecuatia~cu~4~si~grupam~convenabil~termenii: \\ \\ (4x^2-20x+25)+(4y^2-12y+9)=2 \Leftrightarrow (2x-5)^2+(2y-3)^2=2. \\ \\ Dar~2x-5~si~2y-3~sunt~intregi \Rightarrow~(2x-5)^2=1~si~(2y-3)^2=1. \\ \\ Avem~x \in \{2;3 \}~si~y \in \{1;2 \}. \\ \\ Solutie:~ (x,y) = \{2;3\} \times \{1;2 \}~(produs~cartezian). \\ \\ In~tipul~acesta~de~probleme~de~multe~ori~se~incearca~o~rescriere~cu~o \\ \\ suma~de~patrate,~iar~cand~patratele~nu~vin~direct.$

$(x^2-5x-este~un~"fragment",~iar~-5~este~impar)~se~poate~inmulti \\ \\ cu~4.$