Question

Fie numerele a, b, c. Stiind ca $a^{2}*b^{2} +4= a^{2}+3*a*b$ si  $b^{2}* c^{2}+4= b^{2}+3*b*c$ si $c^{2}* a^{2}+4= c^{2}+3*a*c$, aratati ca a=b=c. Va rog de clasa a VII-a rezolvarea. Multumesc!

Answer 1

(ab-2)²=(ab)²-4ab+2²=a²-ab≥0 (1)
(bc-2)²=(bc)²-4ab+2²=b²-bc≥0 (2)
(ac-2)²=(ac)²-4ab+2²=a²-ac≥0 (3)
Adunand relatiile (1)+(2)+(3)=>
a²-ab+b²-bc+a²-ac≥0|·2
2a²-2ab+2b²-2bc+2a²-2ac≥0
a²-2ab+b²+b²-2bc+c²+a²-2ac+c²≥0
(a-b)²+(b-c)²+(a-c)²≥0
a-b=0=>a=c
b-c=0=>b=c
a-c=0=>a=c
a=b=c