If a = 9-4√5. Find √a - 1/√a
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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 ☺️✌️
√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 ☺️✌️
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