Question

If a=3+2√5, then find value of a+1/a

Answer 1

Answer:

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Answer 2

Step-by-step explanation:

Long method:

As a = 3+2√5

So, 1/a = 1/2+2√5

Putting values

a+1/a

3+2√5 + 1/3+2√5

Adding them by applying L. C. M.

(3+2√5 X 3+2√5) + 1 / 3+2√5

(3+2√5)² +1 / 3+2√5

Now applying identity

(a+b)² = a² + b² + 2ab

here a is 3 and b is 2√5

So,

(9 + 20 + 12√5) + 1 / 3+2√5

29 + 12√5 + 1 / 3+2√5

30 + 12√5 / 3+2√5

After simplification

Short method:

Or you can do it by rationalisation of denominator

1/a = 1/ 3+2√5

1/3+2√5 X 3-2√5/3-2√5

3+2√5/ (3+2√5)²

3+2√5/ (3 +2√5) X (3+2√5)

Using identity a² – b² = (a + b)(a – b)

3+2√5/(3)² - (2√5)²

3+2√5/ 9 - 20

3+2√5/ -11

Now adding both

3+2√5 + 3+2√5/-11