Question

Решите пожалуйста 1 пример!! Даю 20 баллов!
$sqrt{x+2sqrt{x-1}} - sqrt{x-2sqrt{x-1}};$

Answer 1

$sqrt{x+2sqrt{x-1}} -sqrt{x-2sqrt{x-1}}=sqrt{x-1+2sqrt{x-1}+1} -\\-sqrt{x-1-2sqrt{x-1}+1}= sqrt{(sqrt{x-1})^2+2cdotsqrt{x-1}cdot 1+1^2} -\ \ - sqrt{(sqrt{x-1})^2-2cdotsqrt{x-1}cdot 1+1^2}=sqrt{(sqrt{x-1}+1)^2} -\\ -sqrt{(sqrt{x-1}-1)^2} =|{sqrt{x-1}+1}|-|{sqrt{x-1}-1}|=\\=sqrt{x-1}+1-|{sqrt{x-1}-1}|=left { {{sqrt{x-1}+1-(sqrt{x-1}-1), sqrt{x-1}-1geq0} atop {{sqrt{x-1}+1-(1-sqrt{x-1}), sqrt{x-1}-1<0}} right. =$

$=left { {{sqrt{x-1}+1-sqrt{x-1}+1, sqrt{x-1}geq1} atop {{sqrt{x-1}+1-1+sqrt{x-1}, sqrt{x-1}<1}} right.=left { {{2, x-1geq1} atop {{2sqrt{x-1}, 0leq x-1<1}} right. =left { {{2, xgeq2} atop {{2sqrt{x-1}, 1leq x<2}} right.$