Given a cubic equation of the form ax

Another quick, but not so comprehensive, test is to look at b

So, in your F(x) = 10x

^{3}+bx^{2}+cx+d=0, an annoyingly complicated formula for the "discriminant" will tell you how many real roots the equation has. The discriminant is 18abcd-4(ac^{3}+b^{3}d)+b^{2}c^{2}-27a^{2}d^{2}. If it is <0, there are two complex roots and one real root. If it is =0, all roots are real and at least two of them are equal. If the discriminant is >0, there are three unequal real roots.Another quick, but not so comprehensive, test is to look at b

^{2}-3ac. If this value is <0, the equation has only one real root. If this value is =0 or >0, something can be said about the number of inflection points in the graph of ax^{3}+bx^{2}+cx+d, but that is all.So, in your F(x) = 10x

^{3}- 4x^{2}+ 2x - 6, the value of b^{2}-3ac is (-4)^{2}-3(10)(2)=16-60=-44. Thus, the function F(x) is monotonic (has no inflection points) and there is only one real root. The other two roots must be complex, i.e. They have a non-zero imaginary part.