Question

Matematica de clasa a 7 a

Answer 1

a)

$\sqrt{4+2\sqrt3}=\sqrt{3+1+2\sqrt3}=\sqrt{(\sqrt3)^2+2\sqrt3+1^2}=\sqrt{(\sqrt3+1)^2}=\sqrt3+1\\ \\ \sqrt{5-2\sqrt6}=\sqrt{3+2+2\sqrt6}=\sqrt{(\sqrt3)^2+2\sqrt3+(\sqrt2)^2}=\sqrt3+\sqrt2\\ \\ |\sqrt2-2|=|\underbrace{2-\sqrt2}_{>0}|=2-\sqrt2$

$\it a=\sqrt3+1-(\sqrt3-\sqrt2)+2-\sqrt2=\sqrt3+1-\sqrt3+\sqrt2+2-\sqrt2=3$

b)

$\it a=3 \Rightarrow \dfrac{a}{x-2}=\dfrac{3}{x-2}, \ \ x\in\mathbb{Z}\\ \\ \\ \dfrac{3}{x-2}\in\mathbb{Z} \Rightarrow x-2\in D_3 \Rightarrow x-2\in\{-3,\ -1,\ 1,\ 3\}|_{+2} \Rightarrow x\in\{-1,\ 1,\ 3,\ 5\}$