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Problem: (4x+1)/(4x-1)+(4x-1)/(4x+1)=3 [ Calculator, Method and examples ]

Solution:
Your problem `->` (4x+1)/(4x-1)+(4x-1)/(4x+1)=3


`(4x+1)/(4x-1)+(4x-1)/(4x+1)=3`

`((4*x+1)/(4*x-1))+((4*x-1)/(4*x+1))=3`

Let `(4*x+1)/(4*x-1)=m`

`=>m+1/m=3`

`=>m^2-3m+1=0`

`=>m^2-3m+1 = 0`

factor is not possible for equation `m^2-3m+1=0`

Solution is possible using the method of perfect square.
Comparing the given equation with the standard quadratic equation `ax^2+bx+c=0,`

we get, `a=1, b=-3, c=1.`

`:. Delta=b^2-4ac`

`=(-3)^2-4 (1) (1)`

`=9-4`

`=5`

`:. sqrt(Delta)=sqrt(5)`



Now, `alpha=(-b+sqrt(Delta))/(2a)`

`=(-(-3)+sqrt(5))/(2*1)`

`=(3+sqrt(5))/2`



and, `beta=(-b-sqrt(Delta))/(2a)`

`=(-(-3)-sqrt(5))/(2*1)`

`=(3-sqrt(5))/2`

Now, `(4*x+1)/(4*x-1)=(3+sqrt(5))/2`

`=>(4*x+1)/(4*x-1)=2.62`

`=>(4*x+1)=2.62(4*x-1)`

`=>4x+1=10.47x-2.62`

`=>4x-10.47x=-1-2.62`

`=>6.47x = 3.62`

`=>x = 3.62/6.47`

`=>x = 0.56`

Now, `(4*x+1)/(4*x-1)=(3-sqrt(5))/2`

`=>(4*x+1)/(4*x-1)=0.38`

`=>(4*x+1)=0.38(4*x-1)`

`=>4x+1=1.53x-0.38`

`=>4x-1.53x=-1-0.38`

`=>2.47x = -1.38`

`=>x = -1.38/2.47`

`=>x = -0.56`






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