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For  x > 1, simplify sqrt(x + sqrt(2x - 1)) - sqrt(x - sqrt(2x - 1)).

 Dec 11, 2019
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For
x > 1, simplify \(\sqrt{ x + \sqrt{2x - 1} } - \sqrt{ x - \sqrt{2x - 1} }\).

 

\(\text{Let $k= \sqrt{2x - 1},$ or $k^2=2x-1$}\)

 

\(\begin{array}{|rcll|} \hline && \mathbf{ \sqrt{ x + \sqrt{2x - 1} } - \sqrt{ x - \sqrt{2x - 1} } } \\\\ &=& \sqrt{ x + k } - \sqrt{ x - k } \\\\ &=& \sqrt{ \left( \sqrt{ x + k } - \sqrt{ x - k } \right)^2 } \\\\ &=& \sqrt{ (x+k) -2\sqrt{ (x+k)(x-k) } + (x-k) } \\\\ &=& \sqrt{ 2x -2\sqrt{ x^2-k^2 } } \quad | \quad k^2=2x-1 \\\\ &=& \sqrt{ 2x -2\sqrt{ x^2-(2x-1) } } \\\\ &=& \sqrt{ 2x -2\sqrt{ x^2-2x+1 } } \\\\ &=& \sqrt{ 2x -2\sqrt{ (x-1)^2 } } \\\\ &=& \sqrt{ 2x -2 (x-1) } \\\\ &=& \sqrt{ 2x -2x +2 } \\\\ &=& \mathbf{ \sqrt{ 2 } } \quad | \quad x \geq 1 \\ \hline \end{array}\)

 

laugh

 Dec 11, 2019

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