Question

Fie x, y, z trei numere reale astefel incat x²-(y-z)²= 10³ si 2z - 2x - 2y = -20 .Aratati ca numarul x - y + z este egal cu patratul uni numar natural.
Repede!

Answer 1

$\it x^2-(y-z)^2 =10^3 \Rightarrow (x-y+z)(x+y-z) =1000\ \ \ \ (1)\\ \\ 2z-2x-2y=-20 \Rightarrow -2(x+y-z)=-20|_{:(-2)} \Rightarrow x+y-z=10\ \ \ \ (2)\\ \\ (1),(2) \Rightarrow (x-y+z)\cdot10=1000|_{:10} \Rightarrow x-y+z=100 \Rightarrow\\ \\ \Rightarrow x-y+z=10^2\ (p\breve{a}trat\ perfect)$