12. If x = 5 + 2root6 and y = 1/x, then find the value of x^2 + y^2.
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x = 5 + 2√6 , y = 1/x
X² + y ² = (5 + 2√6 )² + [1/(5 + 2√6)]²
= 25 + 24 + 20√6 + 1/(25 + 24 + 20√6)
= 49 + 20√6 + 1/( 49 + 20√6)
= [(49 + 20√6)² + 1]/ 49 + 20√6
= [2401 + 2400 + 980√6 + 1]/49 + 20√6
= [(4802 + 980√6)/49 + 20√6] × [(49 - 20√6)/(49 - 20√6)]
= [(4802 + 980√6)(49 - 20√6)]/2401 - 2400
= [4802(49 - 20√6) + 980√6(49 - 20√6)]/1
= 235298 - 96040√6 + 48020√6 - 117600
= 117698 - 48020√6
= 49( 2402 - 980√6)
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