Question

√а+2√а+3+4+√а-2√а+3+4=

Answer 1

$\sqrt{a+2\sqrt{a+3}+4}+\sqrt{a-2\sqrt{a+3}+4}=\\\\=\sqrt{(\sqrt{a+3}+1)^2}+\sqrt{(\sqrt{a+3}-1)^2}=\\\\=|\underbrace {\sqrt{a+3}+1}_{>0}|+|\sqrt{a+3}-1|=\sqrt{a+3}+1+|\sqrt{a+3}-1|=\\\\=\left \{ {{2\sqrt{a+3}\; ,\; esli\; a\geq -2\; ,} \atop {2\; ,\; esli\; a<-2\; .\qquad \; \; }} \right.$

$\star \; \; |\sqrt{a+3}-1|=\sqrt{a+3}-1\; ,\; \; esli\; \sqrt{a+3}-1\geq 0\; ,\; \sqrt{a+3}\geq 1\; \to \\\\a+3\geq 1\; ,\; a\geq -2\; \; \star \\\\\star |\sqrt{a+3}-1|=1-\sqrt{a+3}\; ,\; esli\; \sqrt{a+3}-1<0\; ,\; \sqrt{a+3}<1\; \to \\\\a+3<1\; ,\; a<-2\; \; \star$