If , are the roots of the equations x²-2 x-1=0, then what is the value of ² ⁻²+ ⁻² ²

Step-by-step solution

Since, ( ) and ( ) are the roots of the equation (x²-2 x-1=0 ), then Sum of roots, ( + =2 ) and product of the roots ( =-1 ) Since, (( + )= ²+ ²+2 ) ( 4= ²+ ²-2 ) ( ²+ ²=6 ) Now, ( ² ⁻²+ ⁻² ²= ² ² + ² ² = ⁴+ ⁴ ( )² ) ( ( ²+ ² )²=6² ) ( ⁴+ ⁴+2 ² ²=36 ) ( ⁴+ ⁴+2=36 ) ( ⁴+ ⁴=34 . .( i ) ) ( ⁴+ ⁴ ( )² = 34 (-1)² =34 ) [Putting value of ( ⁴+ ⁴=34 ) from Equation (i) (] )