find the value of x^2+1/x^2 and x^4+1/x^4, if x - 1/x = 3
Answers
First, lets rearrange the equation into quadratic form.
x+1x=3
x2+1=3x
x2−3x+1=0
using the quadratic formula
x=32±9−4√2
case 1—assume that it is the positive square root:
x=32+5√2
you could use binomial theorem if you wanted to calculate it by hand:
x4=8116+42785√2+(42)9454+43255√8+2516
but lets just calculate (32+5√2)4 with a calculator.
x4=46.9787
now lets assume case 2—a negative square root:
x=32−5√2
x4=(32−5√2)4=0.0213
in case 1:
substitute 46.9787 into x4+1x4
46.9787+146.9787=47
in case 2:
0.0213+10.0213=47
interestingly, both solutions to the quadratic yielded the same result because the solutions to this this quadratic were reciprocals:
32+5√2=(32−5√2)−1
3+5√2=23−5√⋅3+5√3+5√
3+5√2=6+25√9−5
3+5√2=6+25√4
3+5√2=3+5√2
This makes sense because the quadratic x2−3x+1=0 has 2 irrational roots, and it can be factored into:
(x−3+5√2)(x−3−5√2)
Because its expanded form has a constant of 1, the product of the roots must be 1. Not only are the roots reciprocals, they are also conjugates. This will be true for any value of b greater than 2 in a quadratic of form x2−bx+1 How cool is that‽