if x^3+1/x^3=52 then find x+1/x
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Hey !!
Let X + 1/X = Y
X³ + 1/X³ = 52
( X + 1/X)³ - 3 ( X + 1/X ) = 52
( Y)³ - 3Y = 52
Y³ - 3Y - 52 = 0
Y³ - 4Y + 4Y - 16Y + 13Y - 52 = 0
Y² ( Y - 4 ) + 4Y ( Y - 4 ) + 13 ( Y - 4 ) = 0
( Y - 4 ) ( Y² + 4Y + 13 ) = 0
( Y - 4 ) = 0
Y = 4.
Hence,
X + 1/X = Y = 4
Let X + 1/X = Y
X³ + 1/X³ = 52
( X + 1/X)³ - 3 ( X + 1/X ) = 52
( Y)³ - 3Y = 52
Y³ - 3Y - 52 = 0
Y³ - 4Y + 4Y - 16Y + 13Y - 52 = 0
Y² ( Y - 4 ) + 4Y ( Y - 4 ) + 13 ( Y - 4 ) = 0
( Y - 4 ) ( Y² + 4Y + 13 ) = 0
( Y - 4 ) = 0
Y = 4.
Hence,
X + 1/X = Y = 4
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