Question

Find the value of x² - x+1 when x=√3 -1, in the form a+b√3 where a and b are integers​

Answer 1

`6-3√(3)`

If you compare this to
`a+b√(3)`, then
`a=6` and
`b=-3`.

Explanation:

We are given
`x=√(3)-1` and we are asked to find the value of
`x^2-x+1`.

So we will need to find
`x^2`.

`x=√(3)-1`

Square both sides:

`x^2=(√(3)-1)^2`

Expand using identity:
`(x-a)^2=x^2-2ax+a^2`.

`x^2=(√(3))^2-2(√(3))(1)+1^2`

`x^2=3-2√(3)+1`

`x^2=4-2√(3)`

Let's go back to the full
`x^2-x+1`.

`x^2-x+1`

`(4-2√(3))-(√(3)-1)+1`

`4-2√(3)-√(3)+1+1`

`6-3√(3)`

If you compare this to
`a+b√(3)`, then
`a=6` and
`b=-3`.