If x=3+2 root 2 then find [root x]-1/[root x]
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I'm assuming your question in 2 ways, as I'm not sure what exactly it is.
x = 3+2√2
√x = √(3+2√2)
= √(2+1+2√2)
= √( (√2)^2+1^2+2×√2×1)
(We get the (a+b)^2 identity here, = a^2+b^2+2×a×b)
= √ (√2+1)^2
= √2+1
1. √x- (1/√x)
= √2+ 1 - (1/√2+1)
= √2+1 - (1/√2+1 × √2-1/√2-1) ( rationalizing)
= √2+1 - (√2- 1)
= √2 + 1 - √2+1= 2 (Ans)
Or.
2. (√x-1)/ √x
= √2+1-1/ √2+1
= √2/ √2+1 × √2-1/√2-1
= 2 - √2/ 2 - 1
= 2- √2 (Ans)
x = 3+2√2
√x = √(3+2√2)
= √(2+1+2√2)
= √( (√2)^2+1^2+2×√2×1)
(We get the (a+b)^2 identity here, = a^2+b^2+2×a×b)
= √ (√2+1)^2
= √2+1
1. √x- (1/√x)
= √2+ 1 - (1/√2+1)
= √2+1 - (1/√2+1 × √2-1/√2-1) ( rationalizing)
= √2+1 - (√2- 1)
= √2 + 1 - √2+1= 2 (Ans)
Or.
2. (√x-1)/ √x
= √2+1-1/ √2+1
= √2/ √2+1 × √2-1/√2-1
= 2 - √2/ 2 - 1
= 2- √2 (Ans)
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