Determine all possible integer values of N that satisfy this equation:
(√5 + 2)
N + (√5 - 2)
N= 18
This equation is of the form:
x^n + (1/x)^n = 18
The 2 conjugates in this case are reciprocals of each other since (√5)^2 - (2)^2 = 1
In this case, x + 1/x = 2√5
(2√5)^2 = 20
(x + 1/x)^2 = x^2 + 2 + (1/x)^2
so
(√5 + 2)^2 + (√5 - 2)^2= 18
And 2 is the only positive solution because the LHS is an increasing function for N>0 and RHS is constant.
However since negative exponents are reciprocals, N = -2 also works.
Solution: {-2,2}
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Posted by Larry
on 2023-10-02 10:26:47 |
Solution
