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

\(\Leftrightarrow x=\sqrt{5+\sqrt{13+x}}\) (\(x\ge0\))

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

\(\Leftrightarrow x^2-9=\sqrt{13+x}-4\)

\(\Leftrightarrow\left(x-3\right).\left(x+3\right)=\dfrac{x-3}{\sqrt{13+x}+4}\)

\(\Leftrightarrow\left[{}\begin{matrix}x=3\\x+3=\dfrac{1}{\sqrt{x+13}+4}\left(∗\right)\end{matrix}\right.\)

Xét (*) ta có VT \(\ge3\) (1)

mà \(VP=\dfrac{1}{\sqrt{x+13}+4}\le\dfrac{1}{4}\) (2)

Từ (1) và (2) dễ thấy (*) vô nghiệm 

Hay x = 3

\(x=\sqrt[3]{5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}}\)

\(\Rightarrow x^3=5+2\sqrt{13}+5-2\sqrt{13}+3\sqrt[3]{\left(5+2\sqrt{13}\right)\left(5-2\sqrt{13}\right)}.x\)

          \(=10+3x\sqrt[3]{25-52}\)

          \(=10+3x\sqrt[3]{-27}\)

           \(=10-9x\)

\(\Rightarrow x^3+9x-10=0\)

\(\Leftrightarrow x^3-x+10x-10=0\)

\(\Leftrightarrow x\left(x^2-1\right)+10\left(x-1\right)=0\)

\(\Leftrightarrow x\left(x-1\right)\left(x+1\right)+10\left(x-1\right)=0\)

\(\Leftrightarrow\left(x-1\right)\left(x^2+x+10\right)=0\)

Vì \(x^2+x+10=\left(x+\frac{1}{2}\right)^2+\frac{39}{4}>0\forall x\)

=> x - 1 = 0

=> x = 1

Thay vào A = 1²⁰¹⁵ - 1²⁰¹⁶ = 0

Vậy A = 0

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

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

\(\Leftrightarrow\left(x^2-5\right)^2=13+x\)

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

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

do x>2 nen x=3

online ngu 

Bạn Tuyển Trần Thị cho mình hỏi là x > 2 ở đâu vậy?

dk  \(x>2\)

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

    \(\left(x^2-5\right)^2=13+x\)

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

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

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

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

do đó \(x-3=0\Leftrightarrow x=3\)

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

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

\(\Rightarrow x^4=25+10\sqrt{13+\sqrt{5+\sqrt{13+\sqrt{5+....}}}}+13+\sqrt{5+\sqrt{13+\sqrt{5+...}}}\)

\(\Leftrightarrow x^4=38+10x^2+x\)

\(\Leftrightarrow x^4-10x^2-x-38=0\)

giải ra tìm x xong

x= √5+√13+√5+√13=√5+√13+√5+√16=

 = √5+√13+√5+4=√5+√13+√9=√5+√13+3

x= \(\sqrt{5+\sqrt{13+\sqrt{5+\sqrt{13}}}}=\sqrt{5+\sqrt{13+\sqrt{5+\sqrt{16}}}}=\)

 = \(\sqrt{5+\sqrt{13+\sqrt{5+4}}}=\sqrt{5+\sqrt{13+\sqrt{9}}}=\)\(\sqrt{5+\sqrt{13+3}}\)

= \(\sqrt{5+\sqrt{16}}=\sqrt{5+4}=\sqrt{9}=3\)

Dễ dàng nhận thấy \(x>0\)

a/ \(x^2=6+\sqrt{6+\sqrt{6+...}}\)

\(\Leftrightarrow x^2=6+x\)

\(\Leftrightarrow x^2-x-6=0\)

\(\Leftrightarrow\left(x-3\right)\left(x+2\right)=0\)

\(\Leftrightarrow x=3\)

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

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

\(\Leftrightarrow x^2-5=\sqrt{x+13}\) (\(x\ge\sqrt{5}\))

\(\Leftrightarrow\left(x^2-5\right)^2=x+13\)

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

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

Do \(x\ge\sqrt{5}\Rightarrow\left\{{}\begin{matrix}x-1>0\Rightarrow x^3-x=x^2\left(x-1\right)>0\\x^2\ge5\Rightarrow3x^2-4>0\end{matrix}\right.\)

\(\Rightarrow x^3+3x^2-x-4>0\)

\(\Rightarrow x=3\)