Question

If x = √3 +i and y= V3 -i,
then find the value of 7x2 - 11xy + 7y2​

Answer 1

$x = \sqrt{3 } + i \\ {x}^{2} = {( \sqrt{3} + i) }^{2} = 3 + {i}^{2} + 2 \sqrt{3} i \\ {x}^{2} = 3 - 1 + 2 \sqrt{3} i \: \: \: \: \: \: \: \: \: ...( {i}^{2} = - 1) \\ {x}^{2} = 2 + 2 \sqrt{3} i \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ...(i) \\ \\ y = \sqrt{3 } - i \\ {y}^{2} = {( \sqrt{3} - i)}^{2} = 3 + {i}^{2} - 2 \sqrt{3} i \\ {y}^{2} = 3 - 1 - 2 \sqrt{3} i \: \: \: \: \: \: \: \: \: \: ...( {i}^{2} = - 1) \\ {y}^{2} = 2 - 2 \sqrt{3} i \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ...(ii) \\ \\ xy = ( \sqrt{3} + i)( \sqrt{3} - i) \\ xy = {( \sqrt{3}) }^{2} - {(i)}^{2} = 3 - ( - 1) = 3 + 1 \\ xy = 4$

Hence,

7x² - 11xy + 7y²

= 7(2 + 2√3i) - 11(4) + 7(2 - 2√3i)

= 14 + 14√3i - 44 + 14 - 14√3i

= 14 + 14 - 44 + 14√3i - 14√3i = 28 - 44 + 0

= -16

Hence, 7x² - 11xy + 7y² = -16.