Question

if x is equal to 9 + 4 root 5 find the value of root x minus 1 / root x​

Answer 1

If x = 9 + 4√5, find the value of √x - 1/√x

(√x - 1/√x)^2 = (√x - 1/√x)(√x - 1/√x) = x - 1 - 1 + 1/x 
= x + 1/x - 2

= (9 + 4√5) + 1/(9 + 4√5) - 2

Rationalize the second term 1/(9 + 4√5):
1/(9 + 4√5) = (9 - 4√5) / [(9 + 4√5)(9 - 4√5)] 
= (9 - 4√5) / (81 - 36√5 + 36√5 - 16*5)
= (9 - 4√5) / (81 - 80)
= (9 - 4√5) / 1
= 9 - 4√5

So we have
(√x - 1/√x)^2 = (9 + 4√5) + (9 - 4√5) - 2
= 9 + 9 + 4√5 - 4√5 - 2
= 18 - 2
= 16

So (√x - 1/√x)^2 = 16
=> √x - 1/√x = √16 = ± 4
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