If and are the roots of the equation x^2+x+1=0 , then the quadratic equation whose roots are ²⁰²³ and ¹⁰¹² is

Step-by-step solution

Given : and are roots of the equation x^2+x+1=0 Then x= -1 1-4 2 = -1 3 i 2 = -1+ 3 i 2 and = -1- 3 i 2 =- 1 2 +i 3 2 and =- 1 2 - 3 2 i = ( 2 3 +i 2 3 )= (e^ i 2 3 ) = ( 4 3 +i 4 3 )= (e^ i 4 3 ) ²⁰²³= (e^ i 2 3 )²⁰²³=e^ i 4046 3 = 4046 3 +i ( 4046 3 )= (1348 + 2 3 )+i (1348 + 2 3 )= ( 2 3 )+i ( 2 3 ) ²⁰²³=- 1 2 +i 3 2 = Now, ¹⁰¹²= (e^ i 4 3 )¹⁰¹²=e^ i 4048 3 = ( 4048 3 )+i ( 4048 3 )= (1349 + 3 )+i (1349 + 3 )= ( + 3 )+i ( + 3 )= 4 3 +i 4 3 ¹⁰¹²= ²⁰²³= and ¹⁰¹²= The required equation whose root is ²⁰²³ and ¹⁰¹² i