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# find the solutions of equation(x-1/x)1/2 + (1-1/x)1/2 = x

Shashwat Tiwari
20 Points
9 years ago
(x-1/x)1/2 = x-(1-1/x)1/2
squaring both the sides..........
x-1/x = x2 + 1 - 1/x - 2(x2-x)1/2 ...................
x2-x+1=2(x2-x)1/2.....................
now, put x2-x=t.......
t+1=2(t)1/2
again square both sides.............
(t-1)2=0
t=1
Hence, x2-x=1
x=(1±(5)1/2)/2..............
jagdish singh singh
173 Points
9 years ago

$\hspace{-16}\mathbf{\sqrt{x-\frac{1}{x}}+\sqrt{1-\frac{1}{x}}=x}\\\\\\ Let \mathbf{\sqrt{x-\frac{1}{x}}=a} and \mathbf{\sqrt{1-\frac{1}{x}}=b}\\\\\\ \mathbf{a+b=x........................................(1)}\\\\\\ \mathbf{a^2-b^2=x-1\Leftrightarrow (a+b).(a-b)=(x-1)}\\\\\\ \mathbf{a-b=\frac{x-1}{x}..................................(2)}\\\\\\ \mathbf{2a=x+\frac{x-1}{x}\Leftrightarrow =1+a^2\Leftrightarrow }\\\\\\ \mathbf{a^2-2a+1=0\Leftrightarrow (a-1)^2=0\Leftrightarrow a=1}\\\\\\ So \mathbf{\sqrt{x-\frac{1}{x}}=a=1\Leftrightarrow x-\frac{1}{x}=1}\\\\\\ So \mathbf{x=\frac{-1+\pm 5}{2}\Leftrightarrow \boxed{\bold{x=\frac{1+\sqrt{5}}{2}}}}$