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July 23rd, 2015, 01:56 PM  #2 
Senior Member Joined: Apr 2014 From: Greater London, England, UK Posts: 320 Thanks: 156 Math Focus: Abstract algebra 
That is the discriminant of the reduced cubic equation $\displaystyle x^3+px+q\ =\ 0$If the discriminant is positive, the equation has three real roots. If zero, the roots are also real but there will be repeated ones. If the discriminant is negative, there is one real and two complex roots. 
July 23rd, 2015, 02:25 PM  #3 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
For the general cubic $ax^3+bx^2+cx+d$ the discriminant is $$ 27a^2d^2  18abcd + 4ac^3 + 4b^3d  b^2c^2. $$ 

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conditions, cubic, equation, roots 
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