if X 2 - 3x + 1 = 0 then value of X 5 + 1/ X 5 Equal to

Question

^if   X ^{2 _{- 3x + 1 = 0}  then value of _X 5 ₊ 1/ _X 5 Equal to}

Answer 1

There is another way to solve it.

x²-3x+1=0 <---(1). [x²+1=3x]

x⁵+1/x⁵=?

(x+1/x)=(x²+1)/x=|due to (1)|=3x/x=3. <---(2)

 

(x+1/x)⁵=x⁵+5x³+10x+10/x+5/x³+1/x⁵=

       =x⁵+1/x⁵+10(x+1/x)+5(x³+1/x³)

x⁵+1/x⁵=(x+1/x)⁵-10(x+1/x)-5(x³+1/x³)=

      =(x+1/x)⁵-10(x+1/x)-5(x+1/x)(x²+1/x²-2) <---- (3)

 

From (2) we have

(x+1/x)=3

x²+2+1/x²=9

x²+1/x²=7

 

x⁵+1/x⁵=(x+1/x)⁵-10(x+1/x)-5(x+1/x)(x²+1/x²-2)

      = 3⁵-10*3-5*3*5=3(81-10-25)=3*46=138

Check out this solution.

Answer 2

x = [3 +or- sqrt( 9 - 4)] / 2

 

   = [ 3 +or- sqrt(5)]/2

 

  

x^2 =  [ 14 +or- 6*sqrt(5)]/4 = [  7 +or- 3*sqrt(5)]/2

 

x^4 = [ 64 +or- 42*sqrt(5) ]/4 = [ 32 +or- 21*sqrt(5)]/2

 

x^5 =   [3 +or- sqrt(5)][32 +or- 21*sqrt(5)]/4 

 

      = [ 201  +or- 95 * sqrt(5) ]/4

 

 

 

The reciprocal , after rationalizing the denominator is:

   4 * [201 -or+ 95*sqrt(5)]/ [ 201^2 + 95^5*5] =

 

   4*[201 -or+ 95*sqrt(5)]/85526

 

 

The final answer is:

 [ 201 +or- 95 * sqrt(5) ]/4 +  4*[201 -or+ 95*sqrt(5)]/85526

 

 = {85526*[ 201 +or- 95 * sqrt(5) ] + 16*[201 -or+ 95*sqrt(5)] } / 342104

 

 = {17187510 +or- 8123450*sqrt(5)}/342104

 

 You can divide everything by 2 if you wish