Question

If $23=x^4+\frac{1}{x^4}$, then what is the value of $x^2+\frac{1}{x^2}$?

Answer 1

`x^2+(1)/(x^2)=5`

Explanation:

`x^4+(1)/(x^4)=23` is given.

We want to find
`x^2+(1)/(x^2)`.

If we square the value we want to find, we should wind up with some terms of the left hand side of the given.

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

Expand:

`x^4+2x^2(1)/(x^2)+(1)/(x^4)` (We used the identity:
`(x+a)^2=x^2+2xa+a^2)` for expansion).

Simplify this value:

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

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

We are given that the sum of the first two terms is 23.

This means
`(x^2+(1)/(x^2))^2=23+2`.

Let's simplify the right hand side.

`(x^2+(1)/(x^2))^2=25`

Now to find the value we want we must simply take the square root of both sides.

`x^2+(1)/(x^2)=\pm √(25)`

Simplify the right hand side:

`x^2+(1)/(x^2)=\pm 5`

Since
`x^2+(1)/(x^2)` is positive for any real value
`x` (that is not zero), then we can conclude
`x^2+(1)/(x^2)=5`.