Question

if a=9+4root 5 and b=1/a then find the value of a^2+b^2

Answer 1

heya ✋

here's ur answer

touch the picture above

#thankyou

Answer 2

Heya there !!!
Here is the answer you were looking for:

Identities used :

${(x + y)}^{2} = {x}^{2} + {y}^{2} + 2xy \\ {(x - y)}^{2} = {x}^{2} + {y}^{2} - 2xy \\ (x + y)(x - y) = {x}^{2} - {y}^{2}$


Given,
a = 9 + 4√5
b = 1/a

So, putting the value of a we get,

$b = \frac{1}{9 + 4 \sqrt{5} }$

On rationalizing the denominator we get,

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

Now putting the values of a and b in a^2 + b^2 we get,

${a}^{2} + {b}^{2} \\ \\ = {(9 + 4 \sqrt{5} )}^{2} + {(9 - 4 \sqrt{5}) }^{2} \\ \\ = ( {(9)}^{2} + {(4 \sqrt{5}) }^{2} + 2(9)(4 \sqrt{5} )) + ( {(9)}^{2} + {(4 \sqrt{5}) }^{2} - 2(9)(4 \sqrt{5} )) \\ \\ = (81 + 80 + 72 \sqrt{5} ) + (81 + 80 - 72 \sqrt{5} ) \\ \\ = 161 + 72 \sqrt{5} + 161 - 72 \sqrt{5} \\ \\ = 161 + 161 \\ \\ = 322$

Hope this helps!!!

If you have any doubt regarding to my answer, please ask in the comment section or inbox me ^_^

@Mahak24

Thanks...
☺☺