Evaluate the following: (x+ x^2-1 )^6+ (x- x^2-1 )^6 - Vidyadip

Question

Evaluate the following:
$\Big(\text{x}+\sqrt{\text{x}^2-1}\Big)^6+\Big(\text{x}-\sqrt{\text{x}^2-1}\Big)^6$

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Answer

$\Big(\text{x}+\sqrt{\text{x}^2-1}\Big)^6+\Big(\text{x}-\sqrt{\text{x}^2-1}\Big)^6$
$=2\Big[{^6\text{C}}_0\text{x}^6+{^6\text{C}}_2\text{x}^4\big(\sqrt{\text{x}^2-1}\big)^2+{^6\text{C}}_4\text{x}^2\big(\sqrt{\text{x}^2-1}\big)^4+{^6\text{C}}_6\big(\sqrt{\text{x}^2-1}\big)^6\Big]$
$=2\Big[\text{x}^6+15\text{x}^4\big(\text{x}^2-1\big)+15\text{x}^2\big(\text{x}^2-1\big)^2+\big(\text{x}^2-1\big)^3\Big]$
$=2\Big[\text{x}^6+15\text{x}^6-15\text{x}^4+15\text{x}^6+15\text{x}^2-30\text{x}^4+\text{x}^6-1-3\text{x}^4+3\text{x}^2\Big]$
$=64\text{x}^6-96\text{x}^4+36\text{x}^2-2$

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