Question

If x=1/(2-3^1/2)then x^3-2x^2-7x+5

Answer 1

ANSWER:

${x}^{3} - 2 {x}^{2} - 7x + 5 = 3$

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SOLUTION:

$this \: is \: a \: very \: intresting \: question \\ \: which \: can \: be \: solved \: from \: a \: logical \\ \: way \: \\ given \: that \: x = \frac{1}{2 - \sqrt{3} } \\ lets \: rationalize \: this \: \\ = > x = \frac{1}{2 - \sqrt{3} } \times \frac{2 + \sqrt{3} }{2 + \sqrt{3} } \\ = > x = \frac{2 + \sqrt{3} }{ {2}^{2} - { (\sqrt{3} )}^{2} } \\ = >x = \frac{2 + \sqrt{3} }{4 - 3} \\ = > x = 2 + \sqrt{3} \\ send \: the \: two \: other \: side \: \\ = > x - 2 = \sqrt{3} \\ = > {(x - 2)}^{2} = 3 \\ = > {x}^{2} - 4x + 4 = 3 \\ = > {x}^{2} - 4x + 1 = 0 \\ find \: the \: {x}^{3} value \: from \: this \: and \: \\ substitute \: in \: given \: equation \: \\ = > {x}^{2} = 4x - 1 \: \: \: \: \: \: \: (1)\\ = > multiply \: x \: both \: sides \\ = > {x}^{3} = 4 {x}^{2} - x\: \: \: \: \: \: \: (2)$

---------------------------------------------------- $= > {x}^{3} - 2 {x}^{2} - 7x + 5 \\ = >( 4 {x}^{2} - x) - 2 {x}^{2} - 7x + 5 \: \: \: \: \: \: \: from(2)\\ = > 2 {x}^{2} - 8x + 5$

---------------------------------------------------- $taking \: (1)again \: we \: get \\ {x}^{2} - 4x = - 1\: \: \: \: \: \: \: (3)$

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$= > 2 {x}^{2} - 8x + 5 \\ = > 2( {x}^{2} - 4x) + 5 \\ = > 2( - 1) + 5 \: \: \: \: \: \: \: from(3)\\ = > 5 - 2 \\ = > 3$

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hope it helps you