If ( 1 + sqrt 2 ) and ( 1 - sqrt 2 ) are the roots of equation ax^2 + bx - 1 = 0, then find the values of a and b.

Question

Let a1 and a2 be the roots or the equation (x- a)(x - b) = c and c != 0, then what are the roots of the equation (x - a1)(x - a2) + c = 0?

Answer 1

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