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Inequalities Involving Modulus Function

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Question

If $α$ and $β$ are the roots of the equation $x^{2} - 34 x + 1 = 0$ , then find $\sqrt[4]{α} - \sqrt[4]{β}$ .

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Solution

$\begin{matrix} x^{2} - 34 x + 1 = 0 L e t i t^{'} s t w o r o o t b e α a n d β α + β = 34 α β = 1 x = α^{1 / 4} - β^{1 / 4} x^{2} = α^{1 / 2} + β^{1 / 2} - 2 {(α β)}^{1 / 4} x^{2} = α^{1 / 2} + β^{1 / 2} - 2 {(x^{2} + 2)}^{2} = α + β + 2 α^{1 / 2} β^{1 / 2} {(x^{2} + 2)}^{2} = 36 x^{2} + 2 = 6 x^{2} = 4 x = \pm 2 \end{matrix}$

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