Question

Rog ajutor la: daca a=√(1-√3)^2 si b=√(4+2√3) aratati ca
a^2-2ab+b^2=4
Multumesc.

Answer 1

 

$\displaystyle\bf\\a=\sqrt{\Big(1-\sqrt{3}\Big)^2}=\Big|1-\sqrt{3}\Big|=\boxed{\bf\sqrt3-1}\\\\\\b=\sqrt{4+2\sqrt{3}}=\sqrt{3+1+2\sqrt{3}}=\sqrt{(\sqrt3)^2+2\cdot\sqrt{3}\cdot1+1^2}=\\\\~~~=\sqrt{(\sqrt3+1)^2}=\Big|\sqrt3+1\Big|=\boxed{\bf\sqrt3+1}\\\\\\a^2-2ab+b^2=(a-b)^2\\\\a=\sqrt3-1~~~si~~~b=\sqrt3+1\\\\\\(a-b)^2=((\sqrt3-1)-(\sqrt3+1))^2=\\\\=(\underline{\sqrt3}-1-\underline{\sqrt3}-1))^2=(-1-1)^2=(-2)^2=\boxed{\bf4}\\\\\\\implies~~\boxed{\bf~a^2-2ab+b^2=4}$