Question

If a = 9-4√5. Find √a - 1/√a

Answer 1

a = 9 - 4√5

√a = √(9 - 4√5)

Let,

√(9 - 4√5) = √c + √b

=> 9 - 4√5 = c + b + 2√cb

Comparing,

c + b = 9
2√cb= 4√5

=> √cb = 2√5

=> CB = (2√5)²

=> cb = 4 × 5

=> cb = 20

Now we have to guess a number whose sum is ( c + b) is 9 and product (cb) is 20

By trial, we find that
c = 5 and b = 4
as c + b = 5 + 4 = 9 and 5 × 4 = 20

So √(9 - 4√5) = √c + √b

=> √(9 - 4√5) = √5 + √4

=> √(9 - 4√5) = √5 + 2

Now √a = √( 9 - 4√5)

=> √a = √5 + 2

So 1/√a = 1/(√5 + 2)

Rationalising,

1/√a = (√5 - 2)/(√5 + 2)( √5 - 2)

=> 1/√a = (√5 - 2)/ (√5² - 2²)

=> 1/√a = (√5 - 2)/( 5 - 4)

=> 1/√a = √5 - 2

and √a = √5 + 2


=> √a - 1/√a = √5 + 2 - (√5 - 2)

=>>√a - 1/√a = √5 + 2 - √5 + 2

=> √a - 1/√a = 4


So 4 is your answer


Hope it helps dear friend ☺️✌️

Answer 2

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