[tex]x^2+3xy+4y^2= \\ \\ =x^2+2 \cdot x \cdot \frac{3y}{2}+ \frac{9y^2}{4}+ \frac{7y^2}{4}= \\ \\ = \underbrace{(x+ \frac{3y}{2})^2} \underbrace{+ \frac{7y^2}{4} } \\~~~~~~~ \geq 0 ~~~~~ \geq 0 \Longrightarrow x^2+3xy+4y^2 \geq 0. \\ \\ Egalitatea~are~loc~daca~ \frac{7y^2}{4}=(x+ \frac{3y}{2})^2=0,~adica~x=y=0. [/tex]
