Question

Question #4824d

Answer 1

`x^2 + 16/x^2 = 28`

Explanation:

The first thing to notice here is that

`x^2 + 16/x^2`

can be written as

`(x + 4/x)^2 - 8`

This is the case because

`(x + 4/x)^2 = x^2 + 2 * color(red)(cancel(color(black)(x))) * 4/color(red)(cancel(color(black)(x))) + (4/x)^2`

`= x^2 + 8 + 16/x^2`

So if you subtract `8` from both sides of this equation, you will end up with

`(x + 4/x)^2 - 8 = x^2 + color(red)(cancel(color(black)(8))) + 16/x^2 - color(red)(cancel(color(black)(8)))`

which is

`(x + 4/x)^2-8 = x^2 + 16/x^2" "color(blue)("(*)")`

Now, you know that

`x = 3 - sqrt(5)`

This means that you have

`x + 4/x = 3 - sqrt(5) + 4/(3 - sqrt(5))`

If you rationalize the denominator of `4/(3 - sqrt(5))` by multiplying both the numerator and the denominator by `3 + sqrt(5)`, the conjugate of `3 - sqrt(5)`, you will end up with

`3 - sqrt(5) + (4 * (3 + sqrt(5)))/((3 - sqrt(5))(3 + sqrt(5))`

At this point, you can use the fact that

`color(blue)(ul(color(black)(a^2 - b^2 = (a-b)(a + b))))`

to say that

`(3 - sqrt(5))(3 + sqrt(5)) = 3^2 - (sqrt(5))^2`

`= 9 - 5`

`= 4`

This means that you have

`3 - sqrt(5) + (color(red)(cancel(color(black)(4))) * (3 + sqrt(5)))/color(red)(cancel(color(black)(4))) = 3 - color(red)(cancel(color(black)(sqrt(5)))) + 3 + color(red)(cancel(color(black)(sqrt(5))))`

`= 6`

Therefore, you can say that

`x + 4/x = 6`

Plug this back into equation `color(blue)("(*)")` to get

`(x + 4/x)^2 - 8 = 6^2 - 8 = 28`

This means that you have

`x^2 + 16/x^2 = 28`