If x + 1 = x2 and x > 0, then 2x4 is:
Explanation:
x + 1 = x2 ⇒ x2 - x – 1 = 0 ⇒ x = (1+√5)/2 (∵ x > 0)
Now, squaring the original equation, we get
(x +1)2 = x4 ⇒ x2 + 1 + 2x = x4 ⇒ x4 = x + 1 + 1 + 2x (Since, x2 = x + 1) ⇒ x4 = 3x + 2 ⇒ 2x4 = 6x + 4 ⇒ 2x4 = 6((1+√5)/2) + 4 ⇒ 2x4 = 7 + 3√5
Alternately, We know, x = (1+√5)/2 ⇒ x ≈ 1.62 ∴ 2x4 ≈ 13.8
option I ≈ 14.8, option II ≈ 11.6, option IV ≈13.6
Hence, option (d).
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