Question

sa se determine nr reale x si y daca x^2+6x+y^2+10y+34=0

Answer 1

$\displaystyle\\x^2+6x+y^2+10y+34=0\\\\x^2+6x+y^2+10y+9+25=0\\\\x^2+6x+9+y^2+10y+25=0\\\\(x^2+2\cdot x\cdot 3 + 3^2) + (y^2 + 2 \cdot y \cdot 5 + 5^2)=0\\\\\Big(x+3\Big)^2+\Big(y+5\Big)^2=0\\\\\text{Avem suma de 2 patrate, fiecare fiind mai mari sau egale cu zero.}\\\\\text{Suma lor este egala cu 0 daca fiecare patrat este egal cu 0.}\\\\\Big(x+3\Big)^2=0\\x+3=0\\\boxed{x_1=x_2=-3}\\\\\Big(y+5\Big)^2=0\\y+5=0\\\boxed{y_1=y_2=-5}$