If x + 1/(x + 1) = 6, then determine the value of (x + 2)³ + 1/(x + 2)³.
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Correct answer: A
Explanation
x + 1/(x + 1) = 6
Let – y = x + 1
y + 1/y = 6
y² + 1 = 6y
y² − 6y + 1 = 0
y = 3 ± 2√2
y + 1 = 4 ± 2√2
a³ + b³ = (a + b)³ − 3ab(a + b)
a = y + 1
b = 1/(y + 1)
ab = 1
a + b = (y + 1) + 1/(y + 1)
= [(y + 1)² + 1]/(y + 1)
= (y² + 2y + 2)/(y + 1)
y² = 6y − 1
a + b = (6y − 1 + 2y + 2)/(y + 1)
= (8y + 1)/(y + 1)
Putting y = 3 + 2√2 -
a + b = 8
a³ + b³ = 8³ − 3×1×8
= 512 – 24 = 488