If for z = α + i β , z + 2 = z + 4 ( 1 + i ) , then α + β and α β are the roots of the equation

Step-by-step solution

Given, z = α + i β   &   z + 2 = z + 4 1 + i Now putting the value of z = α + i β in z + 2 = z + 4 1 + i we get, z + 2 = z + 4 1 + i ⇒ α + 2 2 + β 2 = α + 4 + i β + 4 Now on comparing real and imaginary part, we get ( α + 2 ) 2 + β 2 = α + 4 . . . ( i ) and β + 4 = 0 ⇒ β = - 4     . . . ( i i ) Now solving, ( α + 2 ) 2 + 16 = α + 4 ⇒ α 2 + 4 α + 20 = α 2 + 8 α + 16 ⇒ α