Question

If `x^4 + 1/x^4 = 194, "find"  x+ 1/x`

Answer 1

Given: `x^4 + 1/x^4 = 194`

We know that,

`x^4 + 1/x^4 = (x^2 + 1/x^2)^2 - 2`

`194 = (x^2 + 1/x^2)^2 - 2`

`(x^2 + 1/x^2)^2 = 194 + 2`

`(x^2 + 1/x^2)^2 = 196`

∴ `x^2 + 1/x^2 = +-sqrt196`

∴ `x^2 + 1/x^2 = +-14`

Now, relating this expression to `x + 1/x`

`(x + 1/x)^2 = x^2 + 1/x^2 + 2`

`(x + 1/x)^2 = 14 + 2`

`(x + 1/x)^2 = 16`

`x + 1/x = +-sqrt16`

∴ `x + 1/x = +-4`