The quadratic formula of course is:







D =  is the discriminant, and there are three possibilities:



D < 0     ===>  no real roots (no x-axis crossing)

D  = 0    ===>  real root with multiplicity 2 (one x-axis crossing)

D  > 0    ===>  two distinct real roots (two x-axis crossings)








-----------------------

Side-note/if you are curious...



The quadratic formula can be derived from  by completing the square:



     



Divide thru by a, then subtract c/a from both sides: 

    



Complete the square by adding  to both sides:

    



Factor LHS:

    

        

Take square root of both sides (introduces +/- on RHS):

      =  +/- 

      

Subtract  from both sides:

     



Tidy up:

     



     





     

        

 

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