how to factorise?

　
Factorize is something you do to an expression.
However, you have an equation, which we usually solve.

So are you really looking to factor the expression, or is it just a means to the end of solving the equation? Because expression doesn't factor over the rationals, so you might just as well go ahead and solve the equation without bothering to factor first. We can do this by either using quadratic formula or by completing the square. I'll use the latter:

Complete the square:

x² − 2x − 2 = 0
x² − 2x = 2
x² − 2x + 1 = 2 + 1
(x − 1)² = 3
x − 1 = ± √3
x = 1 ± √3


If you really need to factor the expression, this can also be done by completing the square:

x² − 2x − 2
= (x² − 2x + 1) − 1 − 2
= (x − 1)² − 3
= (x − 1)² − (√3)² -----> difference of squared: a² − b² = (a−b)(a+b)
= (x − 1 − √3) (x − 1 + √3)

x^2 - 2x - 2 =0
Does NOT factorise.
Hence to resolve ; 'Complete the Square'.

x^2 - 2x = 2
(x - 1)^2 - 1 = 2
(x - 1)^2 = 3
x - 1 = +/- sqrt(3)
x = 1 +/- sqrt(3)

x² - 2x - 2 = 0

quadratic formula
x = [2 ± √(2² – 4·1(-2))] / [2·1]
 = [2 ± √12] / 2
 = [2 ± 2√3] / 2
 = 1 ± √3
x² - 2x - 2 = (x-1+√3) (x-1-√3)

Quadratic equation

x^2 - 2x - 2 = 0

[-(-2) +/- sqrt((-2)^2 - 4*1*(-2))]/(2*a)
(2 +/- sqrt( 4+8))/2
(2 +/- sqrt(12))/2
(2 +/-2 sqrt(3))/2
1+/- sqrt(3)

(x + (1 + sqrt(3))(x + (1 - sqrt(3)))