Question

x³(x³-18)+1=0
x⁴(x⁴+m)=1
Find m
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Answer 1

Answer:

x³ + 1/x³=18

x³+1/x³=(x+1/x)³-3(x)(1/x)(x+1/x)

18=(x+1/x)³-3(x+1/x)

Let y=x + 1/x

Then:

y³-3y=18

y³-3y-18=0

(y-3)(y²+3y+6)=0

The only real solution to this is y=3; So:

x + 1/x=3 ………………..

x^3 + 1/x^3 = 18

x^6 + 1 = 18x^3

x^6 – 18x^3 + 1 = 0

x^3 = (18 + Sqrt(18^2 - 4))/2 or (18 - Sqrt(18^2 - 4))/2

x^3 = 9 + sqrt(9^2 - 1) or 9 + sqrt(9^2 - 1)

x = (9 + sqrt(9^2 - 1))^(1/3) or (9 - sqrt(9^2 - 1))^(1/3)

x + 1/x = (9 + sqrt(9^2 - 1))^(1/3) + 1/((9 + sqrt(9^2 - 1))^(1/3) ) = 3

or (9 - sqrt(9^2 - 1))^(1/3) + 1/((9 - sqrt(9^2 - 1))^(1/3) ) = 3

Could the last expression be simplified?