Solve the equation:
$x^{4} - 10 x^{3} + 26 x^{2} - 10 x + 1 = 0$ .

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Solution

Given equation, $x^{4} - 10 x^{3} + 26 x^{2} - 10 x + 1 = 0$

$⟹ (x^{2} + x^{- 2}) - 10 (x + x^{- 1}) + 26 = 0 [dividing by x^{2}]$

Substitute $x + x^{- 1} = y$ in the above equation

$⟹ (y^{2} - 2) - 10 y + 26 = 0 ⟹ y^{2} - 10 y + 24 = 0 ⟹ (y - 6) (y - 4) = 0$

$⟹ x + x^{- 1} = y = 4 and 6$

$⟹ x^{2} - 6 x + 1 = 0 and x^{2} - 4 x + 1 = 0$

$⟹ x = 3 \pm 2 \sqrt{2} and x = 2 \pm \sqrt{3}$

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