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Thursday, January 9, 2020

2020/003) If 1 < x < 2, Then Prove that \frac{1}{\sqrt{x+2\sqrt{x-1}}}+ \frac{1}{\sqrt{x-2\sqrt{x-1}}} = \frac {2}{2-x}

Solution

Let \sqrt{x+2\sqrt{x-1}} = \sqrt{m} + \sqrt{n}
Hence  \sqrt{x-2\sqrt{x-1}} = \sqrt{m} - \sqrt{n}
Without loss of generality let us assume m >=n
Square both sides to get  x+2\sqrt{x-1} = m + n + 2\sqrt{mn}
Comparing rational and surds on both sides we get
x=m +n\cdots(1)
x-1= mn\cdots(2)
From (1) and (2)
(m-n)^2 = (m+n)^2-4mn = x^2-4(x-1) = (2-x)^2 we chose 2-x as x < 2
or m-n=2-x\cdots(3)
From (3) and (1) we get m=1, n = x-1
Hence we get
\frac{1}{\sqrt{x+2\sqrt{x-1}}}+ \frac{1}{\sqrt{x-2\sqrt{x-1}}}
= \frac{1}{\sqrt{m}+\sqrt{n}}+ \frac{1}{\sqrt{m}-\sqrt{n}}
= \frac{\sqrt{m}+\sqrt{n}+ \sqrt{m}-\sqrt{n}} {m-n}
= \frac{2\sqrt{m}} {m-n}
= \frac{2\sqrt{1}} {2-x}
= \frac{2} {2-x}




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