Question

Aratati ca,pentru orice x,y apartine R,are loc relatia x la a doua +y la a doua>6x+4y-17

Answer 1

$\displaystyle x^2+y^2\ \textgreater \ 6x+4y-17 \Leftrightarrow \\ \\ \Leftrightarrow x^2+y^2-6x-4y+17\ \textgreater \ 0 \Leftrightarrow \\ \\ \Leftrightarrow \left(x^2-6x+9 \right)+ \left(y^2-4y+4 \right)+4\ \textgreater \ 0 \Leftrightarrow \\ \\ \Leftrightarrow (x-3)^2+(y-2)^2+4\ \textgreater \ 0. \\ \\ Ultima~relatie~este~adevarata~doarece~a^2 \geq0 ~ \forall~a \in \mathbb{R}. \\ \\ \Big( \left(x-3 \right )^2+ \left(y-2 \right)^2+4 \geq 0+0+4=4\ \textgreater \ 0 \Big)$