The value of the expression $x^{4} - 8 x^{3} + 18 x^{2} - 8 x + 2$ when $x = 2 + \sqrt{3}$

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Solution

The correct option is B $1$
$x = 2 + \sqrt{3}$
or ${(x - 2)}^{2} = {(\sqrt{3})}^{2}$
or $x^{2} - 4 x + 1 = 0$
or ${(x - 2)}^{4} = 9$
or $x^{4} - 8 x^{3} + 24 x^{2} - 32 x + 16 = 9$
or $x^{4} - 8 x^{3} + 18 x^{2} - 8 x + 2 + 6 (x^{2} - 4 x + 1) - 1 = 0$
As $x^{2} - 4 x + 1 = 0,$
we get,
$x^{4} - 8 x^{3} + 18 x^{2} - 8 x + 2 = 1$

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x^{4} - 8 x^{3} + 18 x^{2} - 8 x + 2

x = 2 + \sqrt{3}

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