If a= 2 7 +i 2 7 , then the quadratic equation whose roots are =a+a^2+a^4 and =a^3+a^5+a^6 , is

Step-by-step solution

aligned & Given, a= 2 7 +i 2 7 & aligned a^7 & = 2 +i 2 & =1 aligned aligned Also, =a+a^2+a^4, =a^3+a^5+a^6 then the sum of roots, aligned S & = + =a+a^2+a^3+a^4+a^5+a^6 S & = a (1-a^6 ) 1-a = a-a^7 1-a & = a-1 1-a =-1 [ a^7=1 ] aligned Product of the roots, aligned & P= = (a+a^2+a^4 ) (a^3+a^5+a^6 ) & =a^4+a^5+1+a^6+1+a^2+1+a+a^3 [ a^7=1 ] & =3+ (a+a^2+a^3+a^4+a^5+a^6 )=3-1=2 aligned Hence, the required quadratic equation is x^2+x+2=0