SOLUTION: Without solving, determine the character of the solutions of each equation in the complex number system. 1. 2x^2 - 4x + 1 = 0 2. x^2 + 6 = 2x

(1)  For this quadratic equation, calculate the discriminant

         d = b^2 - 4ac.

     Referring to the general form equation, a= 2;  b= -4;  c= 1;  so

         d = (-4)^2 - 4*2*1 = 16 - 8 = 8.


     The discriminant is positive; so this equation has two distinct real solutions.


     At this point, the analysis for part (1) is complete.



(2)  Write the given quadratic equation in standard form

         x^2 - 2x + 6 = 0.

     For this quadratic equation, calculate the discriminant

         d = b^2 - 4ac.

     In this case, a= 1;  b= -2;  c= 6;  so

         d = (-2)^2 - 4*1*6 = 4 - 24 = -20.


     The discriminant is negative; so this equation has no real solutions. 
     It has two distinct solutions in complex numbers.


     At this point, the analysis for part (2) is complete.