"√radical mare t²+2+2√t²+1-√radical mare √t²+2-2√t²+1 egal cu 2"
[tex]\it{\sqrt{t^2+2+2\sqrt{t^2+1}}} -{\sqrt{t^2+2-2\sqrt{t^2+1}}} =2 [/tex]
[tex]\it Notam\ \ t^2+1 = x \Longrightarrow \ x\ \geq \ 1,\ \forall x\in\mathbb{R} [/tex]
[tex]\it Egalitatea \ devine:[/tex]
[tex] \sqrt{x+1+2\sqrt x} -\sqrt{x+1-2\sqrt x} =2\Leftrightarrow \sqrt{(\sqrt x +1)^2} -\sqrt{(\sqrt x -1)^2} =2[/tex]
[tex]\it \Leftrightarrow |\sqrt x+1| - |\sqrt x -1 | =2[/tex]
Deoarece x ≥ 1 ⇒ expresiile din cele două module sunt nenegative, iar ultima egalitate devine:
[tex]\it \sqrt x + 1 -(\sqrt x-1) =2 \Leftrightarrow \sqrt x + 1 -\sqrt x+1 =2 \Leftrightarrow 1+1=2 \Leftrightarrow
\\\;\\
\Leftrightarrow \it2= 2\ (A)[/tex]
