Question

X=root 5+root 4 find x^2+1/x^2

Answer 1

`10-4\sqrt5`

Explanation:

`x=√(5) +√(4)`

Simplify square root 4 to 2:

`x=2+\sqrt5`

Now we can evaluate
`(x^2+1)/(x^2)`. First we can simplify this fraction to
`1 + (1)/(x^2)` and then plug x in.

Plug x in:
`1+(1)/((2+\sqrt5)^2)`. Using
`(x+y)^2=x^2+2xy+y^2`, we can evaluate
`(2+\sqrt5)^2=4+4\sqrt5+5=9+4\sqrt5`.

Now we can rationalize the denominator of the fraction by multiplying by the conjugate term:

`(1)/(9+4\sqrt5) \cdot (9-4\sqrt5)/(9-4\sqrt5) = (9-4\sqrt5)/(81-80)=(9-4\sqrt5)/(1)`

Now we can find the answer:
`1 + 9-4\sqrt5=10-4\sqrt5`