if x - 1/x = 3. find the value of x^2+ 1/x^2 and x^4 + 1/x^4
Answers
Step-by-step explanation:
First, lets rearrange the equation into quadratic form.
x+1x=3x+1x=3
x2+1=3xx2+1=3x
x2−3x+1=0x2−3x+1=0
using the quadratic formula
x=32±9−4√2x=32±9−42
case 1—assume that it is the positive square root:
x=32+5√2x=32+52
you could use binomial theorem if you wanted to calculate it by hand:
x4=8116+42785√2+(42)9454+43255√8+2516x4=8116+427852+(42)9454+432558+2516
but lets just calculate (32+5√2)4(32+52)4 with a calculator.
x4=46.9787x4=46.9787
now lets assume case 2—a negative square root:
x=32−5√2x=32−52
x4=(32−5√2)4=0.0213x4=(32−52)4=0.0213
in case 1:
substitute 46.978746.9787 into x4+1x4x4+1x4
46.9787+146.9787=4746.9787+146.9787=47
in case 2:
0.0213+10.0213=470.0213+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)−132+52=(32−52)−1
3+5√2=23−5√⋅3+5√3+5√3+52=23−5⋅3+53+5
3+5√2=6+25√9−53+52=6+259−5
3+5√2=6+25√43+52=6+254
3+5√2=3+5√23+52=3+52
This makes sense because the quadratic x2−3x+1=0x2−3x+1=0 has 2 irrational roots, and it can be factored into:
(x−3+5√2)(x−3−5√2)(x−3+52)(x−3−52)
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+1x2−bx+1 How cool is that
. Hope it's help
mark as brainlist
sorry l have only 2nd case answer
Nitya❣️
Step-by-step explanation:
I hope it's help you!
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