Question

if a=5-2 root 6 then find a2+1/a2

Answer 1

Hey friend,
Here is the answer you were looking for:
$a = 5 - 2 \sqrt{6} \\ \\ \frac{1}{a} = \frac{1}{5 - 2 \sqrt{6} }$

On rationalizing the denominator we get,

$\frac{1}{a} = \frac{1}{5 - 2 \sqrt{6} } \times \frac{5 + 2 \sqrt{6} }{5 + \sqrt{6} }$

Using the identity :

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

Putting x = 5
and y = 2√6

$= \frac{5 + 2 \sqrt{6} }{ {(5)}^{2} - {(2 \sqrt{6} })^{2} } \\ \\ = \frac{5 + 2 \sqrt{6} }{ {(5)}^{2} - {(2 \sqrt{6} )}^{2} } \\ \\ = \frac{5 + 2 \sqrt{6} }{25 - 24} \\ \\ = 5 + 2 \sqrt{6} \\ \\ a + \frac{1}{a} = (5 - 2 \sqrt{6} ) + (5 + 2 \sqrt{6} ) \\ \\ a + \frac{1}{a} = 5 - 2 \sqrt{6} + 5 + 2 \sqrt{6} \\ \\ a + \frac{1}{a} = 10$

Squaring both the sides we get :

${(a + \frac{1}{a} )}^{2} = {(10)}^{2} \\ \\ {(a)}^{2} + {( \frac{1}{a} )}^{2} + 2 \times a \times \frac{1}{a} = 100 \\ \\ {a}^{2} + \frac{1}{ {a}^{2} } + 2 = 100 \\ \\ {a}^{2} + \frac{1}{ {a}^{2} } = 100 - 2 \\ \\ {a}^{2} + \frac{1}{ {a}^{2} } = 98$


Hope this helps!!

If you have any doubt regarding to my answer, feel free to ask in the comment section or inbox me if needed.

@Mahak24

Thanks...
☺☺

Answer 2

HELLO DEAR,


GIVEN THAT:-


a = (5 - 2√6)
[ on Squaring both side ]

we get,

a² = (25 + 24 - 20√6)

a² = (49 - 20√6)---------------( 1 )


now,

1/a² = 1/(49 - 2√6)


1/a² = 1/(49 - 20√6) * [ (49 + 20√6)/(49 +20√6) ]


1/a² = (49 + 20√6)/( 2401 - 2400)


1/a² = (49 + 20√6)-----------(2)


adding----(1) & -----(2)


we get,


(a² + 1/a²) = (49 - 20√6) + (49 + 20√6)

(a² + 1/a²) = 49 + 49 + 20√6 - 20√6

(a² + 1/a²) = 98



I HOPE ITS HELP YOU DEAR,
THANKS