Question

Cum determin a∈R pentru care expresia: radical din (a²-2a+1)+ radical din (a²+2a+1) are valoare constanta?

Answer 1

La mulţi ani !,

$E=\sqrt{a^2-2a+1}+ \sqrt{a^2+2a+1}=\sqrt{(a-1)^2}+\sqrt{(a+1)^2}=\\=|a-1|+|a+1|.\\\\|a-1|=\begin{cases}\left{a-1,\;dac\u{a}\;a-1\geqslant 0\to a\geqslant1;\\1-a,\;dac\u{a}\;a-1<0\to a<1}.\end{cases}\\\\|a+1|=\begin{cases}\left{a+1,\;dac\u{a}\;a+1\geqslant 0\to a\geqslant-1;\\-a-1,\;dac\u{a}\;a+1<0\to a<-1}.\end{cases}\\\\Pentru\;a\in(-\infty,-1):\;E=1-a-a-1=-2a,\;deci\;E\;NU\;este\;constant;\\\\Pentru\;a\in[-1,1):\;E=1-a+a+1=2,\;deci\;E\;este\;constant;\\\\Pentru\;a\in[1,+\infty):\;E=a-1+a+1=2a,\;deci\;E\;NU\;este\;constant.\\\\Solu\c{t}ia\;este\;a\in[-1,1).$

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