Question

If a=9 - 4√5, find the value of a sq + 1/a sq.​

Answer 1

$= {(9 - 4 \sqrt{5}) }^{2} + \frac{1}{ { (9 - 4 \sqrt{5})}^{2} }$

We first calculate (9-4√5) ^2

$= {(9 - 4 \sqrt{5}) }^{2} \\ = {9}^{2} - 2(9)(4 \sqrt{5} ) + {(4 \sqrt{5}) }^{2} \\ = 81 - 72 \sqrt{5} + 20 \\ = 101 - 72 \sqrt{5}$

Put values:-

$= 101 - 72 \sqrt{5} + \frac{1}{ 101 - 72 \sqrt{5} } \\ = 101 - 72 \sqrt{5} + \frac{1}{ \: 101 - 72 \sqrt{5} } \: \times \frac{101 + 72 \sqrt{5} }{101 + 72 \sqrt{5} } \\ = 101 - 72 \sqrt{5} + \frac{101 + 72 \sqrt{5} }{10201 - 360} \\ = 101 - 72 \sqrt{5} + \frac{101 + 72 \sqrt{5} }{9841} \\ = 101 - 72 \sqrt{5} + 101 + 72 \sqrt{5} - 9841 \\ = 101 - 72 \sqrt{5} + 72 \sqrt{5 } - 9440 \\ = 101 - 9440 \\ = - 9339$

Hope it helps.

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