We will solve the system of two equations as follows:

Let introduce the root values into the equation ax^2 + bx - 1= 0

a( 1+sqrt 2 )^2 + b ( 1 + sqrt 2) - 1 = 0

a( 1 - sqrt 2)^2 + b ( 1 - sqrt 2) - 1 = 0

Solving the system of two equations, yields : a = - b/ 2

Replacing the value of a into the first equation, yields: b = -2 Then a = 1

The cuadratic equation is x^2- 2x -1 = 0