For b denoting sqrt(2) find all possible values of x, satisfying :
(b+1)^x +(b-1)^x=34
>> b=sym(sqrt(2))
b =
2^(1/2)
>> syms x
>> eq=(b+1)^x+(b-1)^x==34
eq =
(2^(1/2) - 1)^x + (2^(1/2) + 1)^x == 34
>> solve(eq,x)
Warning: Unable to solve symbolically.
Returning a numeric solution using
vpasolve.
> In sym/solve (line 304)
ans =
4.0
>>
But this is not complete:
From Wolfram Alpha:
x = (log(17 + 12 sqrt(2)) + 2 i pi n)/log(sqrt(2) - 1) and n an element of Z
x = (-log(17 + 12 sqrt(2)) + 2 i pi n)/log(sqrt(2) - 1) and n an element of Z
where Z is the set of integers
Real solutions:
x = log(17 - 12 sqrt(2))/log(sqrt(2) - 1)
x = log(17 + 12 sqrt(2))/log(sqrt(2) - 1)
Approximate forms:
x˜-1.13459265710651 (3.52549434807817 + 6.28318530717959 i n) and n element Z
x˜-1.13459265710651 (-3.52549434807817 + 6.28318530717959 i n) and n element Z
Real solutions:
x =~ -4
x =~ 4
That's the case of n = 0 in the full list of solutions, both real and complex.
Back to Matlab:
>> clearvars
>> b=sqrt(2); x=-4;
>> (b+1)^x +(b-1)^x
ans =
34
so -4 checks out as the other real solution.
Edited on April 11, 2023, 3:16 pm
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Posted by Charlie
on 2023-04-11 15:13:43 |
all values, via computer
