if x=8-√60 then 1/2[√x+2/√x] is
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x = 8 - √60 = 8 - 2√15 = √3² + √5² -2√3.√5 = (√5 -√3)²
now ,
1/2 [ ( √x + 2)/√x ]
= 1/2[ {√(√5 -√3 )² +2}/√(√5 -√3)² ]
= 1/2[ ( √5 -√3 +2)/(√5 -√3)]
= 1/2 [ ( √5-√3 +2)(√5 +√3)/(√5² -√3²)]
= 1/2[ ( √5² -√3² +2√5 +2√3)/2 ]
= 1/2 ( √5 +√3 +1 )
now ,
1/2 [ ( √x + 2)/√x ]
= 1/2[ {√(√5 -√3 )² +2}/√(√5 -√3)² ]
= 1/2[ ( √5 -√3 +2)/(√5 -√3)]
= 1/2 [ ( √5-√3 +2)(√5 +√3)/(√5² -√3²)]
= 1/2[ ( √5² -√3² +2√5 +2√3)/2 ]
= 1/2 ( √5 +√3 +1 )
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