Question

Tìm x biết

x=\(\sqrt{5+\sqrt{13+\sqrt{5+\sqrt{13+}}}}...\)......

Answer 1

dk  \(x>2\)

Xét \(x^2=5+\sqrt{13+\sqrt{5+\sqrt{13+\sqrt{5+...}}}}\)

    \(\left(x^2-5\right)^2=13+x\)

\(\Leftrightarrow x^4-10x^2-x+12=0\)

\(\Leftrightarrow\left(x^4-9x^2\right)-\left(x^2-9\right)-\left(x-3\right)=0\)

\(\Leftrightarrow\left(x-3\right)\left[\left(x+3\right)\left(x+1\right)\left(x-1\right)-1\right]=0\)

Answer 2

tiếp :  vì \(x>2\Rightarrow\left(x+3\right)\left(x+1\right)\left(x-1\right)-1>0\)

do đó \(x-3=0\Leftrightarrow x=3\)