Question

if a = 9+4√5 and b = 1/a, find a^2 + b^2

Answer 1

Step-by-step explanation:

a = 9+ 4√5

b =1/a

$b = \frac{1}{9 + 4 \sqrt{5} } \\ = \frac{9 - 4 \sqrt{5} }{(9 + 4 \sqrt{5})(9 - 4 \sqrt{5} ) } \\ = \frac{9 - 4 \sqrt{5} }{ {9}^{2} - ( {4 \sqrt{5} })^{2} } \\ = \frac{9 - 4 \sqrt{5} }{81 - 80} \\ = 9 - 4 \sqrt{5}$

Now,

a² + b²= (a + b)² - 2ab

=( 9+4√5+9-4√5)² - 2. [ ab = a* 1/a = 1]

= (18)² - 2

= 324 - 2

= 322

Therefore,

a² + b² = 322

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