Question

If `x = 1/(3 - x),` find the value of `x^3 + 1/x^3`

Answer 1

Given, `x = 1/(3 - x),`

Let’s solve the equation first.

x(3 − x) = 1

3x − x² = 1

x² − 3x + 1 = 0

Now, dividing the whole equation by x:

`x^2/x - (3⁢x)/x + 1/x = 0/x`

`x - 3 + 1/x = 0`

`x + 1/x = 0 + 3`

∴ `x + 1/x = 3`

Using the identity,

a³ + b³ = (a + b)³ − 3ab(a + b)

Here, let a = x and b = `1/x`, also a + b = 3

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

`x^3 + 1/x^3 = 27 - 3(1) (3)`

`x^3 + 1/x^3 = 27 - 9`

∴ `x^3 + 1/x^3 = 18`