China | Can you solve this? | Math

Nice Equation Solving

[TwoMarlboro]

finding the complex roots, realizing the norm is 1 will very quickly lead to an answer. multiply the angle by 29 and -29.

[yurenchu]

Given x + 1/x = √2,
calculate x²⁹ + 1/x²⁹

Note that x=0 is not a solution to the equation; hence x≠0 .

x + 1/x = √2
... square both sides ...
x² + 2 + 1/x² = 2
x² + 1/x² = 0
... multiplying by x² not a problem since x≠0 ...
x⁴ + 1 = 0
x⁴ = -1

Now use this result to evaluate x²⁹ + 1/x²⁹ :

x²⁹ + 1/x²⁹ =
= [x¹ * (x⁴)⁷] + 1/[x¹ * (x⁴)⁷]
= [x * (-1)⁷] + 1/[x * (-1)⁷]
= [x * (-1)] + 1/[x * (-1)]
= -x + 1/(-x)
= -(x + 1/x)
= -√2

[luismuller]

You should probably have mentioned that x is not a real number (because the equation x + 1/x = sqrt(2) can not be true for any real number) but rather a complex number. Assuming there exists a real number x that satisfies this equation, we can multiply with x on both sides and get the equation
x² + 1 = sqrt(2)*x. Obviously (for all real numbers) the left hand side of the equation is positive, so the right hand side should also be positive. Thus we know
sqrt(2)*x > 0. (for x = 0 the expression x + 1/x is undefined). So we can conclude that x > 0. If x > 0 then 1/x > 0 as well. Now we know from the inequality of the arithmetic and geometric mean (you can only use this inequality when a and b are positive) that a+b >= 2*sqrt(a*b). So if we insert x for a and 1/x for b, we get x + 1/x >= 2 * sqrt((x) * (1/x)) = 2. So x + 1/x is always greater than or equal to 2 for all positive real numbers x. So there is no real x for which x + 1/x = sqrt(2). Interestingly, if we assumed that x was a complex number (x = a + bi, then 1/x = (a - bi) / (a² + b²)), then there would be one solution (if x = sqrt(i), then you have a solution for x + 1/x = sqrt(2)). If you now look at the question (x^29 + 1/ (x^29)), you could simply insert sqrt(i) into x and get the value i^(29 / 2) + i ^(-29 / 2) = (i ^ 12) * i^(5 / 2) + i ^ (-12) * i ^ (-5 / 2) = i ^ 2 * sqrt(i) + i ^ (-2) * i^(-1/2) = -(sqrt(i) + 1/sqrt(i)) = -sqrt(2) which is the exact solution you got.

[yadukrishnan]

This is a great explanation thanks alot

[manaskumarchakrabarty]

How can you write x^4+1=0 all of a sudden ? How did you arrive at that from x^2+1/x^2=0 ? Secondly, how an even power 4 of x be a negative number ?

[ashwanibeohar]

Please put "equal to" sign at each step

[bidiptosarkar]

Please solve:
t³-t²-t-287=0(India) please

[Stolz_statt_Scholz]

How can you just make 1/ x^2 to (x^2)+1

[rangarajanvenkatraman]

There seems to be something wrong.If x= _√2, x^4=4, and not -1 as already got.