Question

if x= 1/2+√3 then the value of x cube - x square - 11x + 3​

Answer 1

Step-by-step explanation:

$x = \frac{1}{2 + \sqrt{3} } \\ x = \frac{1}{2 + \sqrt{3} } \times \frac{2 - \sqrt{3} }{2 - \sqrt{3} } \\ x = \frac{2 - \sqrt{3} }{ {2}^{2} - { (\sqrt{3}) }^{2} } \\ x = \frac{2 - \sqrt{3} }{4 - 3} = 2 - \sqrt{3} \\ \\ {x}^{2} = {(2 - \sqrt{3} )}^{2} = {2}^{2} - (2 \times 2 \times \sqrt{3} ) + {( \sqrt{3} )}^{2} \\ {x}^{2} = 4 - 4 \sqrt{3} + 3 = 7 - 4 \sqrt{3}$

${x}^{3} - {x}^{2} - 11x + 3 \\ = x( {x}^{2} - x - 11) + 3 \\ = (2 - \sqrt{3} )(7 - 4 \sqrt{3} - (2 - \sqrt{3} ) - 11) + 3 \\ = (2 - \sqrt{3} )(7 - 4 \sqrt{3} - 2 + \sqrt{3} - 11) + 3 \\ = (2 - \sqrt{3} )( - 6 - 3 \sqrt{3} ) + 3 \\ = - 3(2 - \sqrt{3} )(2 + \sqrt{3} ) + 3 \\ = ( - 3)(2^2 -(\sqrt{3})^2) + 3 \\ = ( - 3)( 4-3) + 3 \\ = ( -3)(1) + 3 = -3+ 3 =0$