Question

Sa se arate ca x²+3xy+4y²≥0, x,y∈R

Answer 1

$x^{2} +3xy+4 y^{2} = \\ \\ = x^{2} +2 \cdot \frac{3}{2} \cdot xy +4 y^{2} = \\ \\ = x^{2} +2 \cdot \frac{3}{2} \cdot xy+ \frac{9}{4}\cdot y^2 + \frac{7}{4} \cdot y^2= \\ \\ =(x+ \frac{3y}{2} )^2+ \frac{7}{4} \cdot y^2 \geq 0+0=0. \\ \\ Egalitate~pentru~x=y=0.$