Question

If `x = 1/(3 - x),` find the value of `x^2 + 1/x^2`.

Answer 1

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

Let’s solve the equation first.

x(3 − x) = 1

3x − x² = 1

x² − 3x + 1 = 0   ...(1)

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`   ...(2)

Squaring both sides of equation (2):

`(x + 1/x)^2 = (3)^2`

`x^2 + 1/x^2 + 2 = 9`

`x^2 + 1/x^2 = 9 - 2`

∴ `x^2 + 1/x^2 = 7`