x + x − 1 = 66 x+x^{-1}=66

Algebra Level 2

If x > 1 x > 1 and x + x − 1 = 66 , x+x^{-1}=66, what is the value of x 1 2 − x − 1 2 ? x^{\frac{1}{2}}-x^{-\frac{1}{2}}?

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Ajay Dutta by Ajay Dutta , 24, India

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2 solutions

Abhishek Singh
Mar 12, 2014

( x 1 2 − x − 1 2 ) 2 = x + x − 1 − 2 × x 1 2 × x − 1 2 (x^{\frac{1}{2}}-x^{-\frac{1}{2}})^{2}= x + x^{-1} -2 \times x^{\frac{1}{2}} \times x^{-\frac{1}{2}} which equals ( x 1 2 − x − 1 2 ) 2 = 66 − 2 (x^{\frac{1}{2}}-x^{-\frac{1}{2}})^{2}=66-2 hence x 1 2 − x − 1 2 = 64 = 8 x^{\frac{1}{2}}-x^{-\frac{1}{2}}=\sqrt{64}=8

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Saurabh Mallik
May 12, 2014

( x + 1 x ) 2 = x + 1 x − 2 (\sqrt{x}+\frac{1}{\sqrt{x}})^{2}=x+\frac{1}{x}-2

( x + 1 x ) 2 = 66 − 2 (\sqrt{x}+\frac{1}{\sqrt{x}})^{2}=66-2

( x + 1 x ) 2 = 64 (\sqrt{x}+\frac{1}{\sqrt{x}})^{2}=64

x + 1 x = 64 \sqrt{x}+\frac{1}{\sqrt{x}}=\sqrt{64}

x + 1 x = 8 \sqrt{x}+\frac{1}{\sqrt{x}}=8

Thus, the answer is: x + 1 x = 8 \sqrt{x}+\frac{1}{\sqrt{x}}=\boxed{8}

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