Need to find parameters for variables 'a' 'b' 'c' that lead to the graph possessing exactly 2 x axis intercepts and for that I need the discriminant of this equation
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Do you mean https://en.wikipedia.org/wiki/Cubic_equation#Discriminant? – Moo Jul 24 '23 at 12:19
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I assume that $a,b,c \in \mathbb{R}$ (you don't say it). In this case, a third degree equation cannot have "just" 2 real roots : if it has 2 real roots, the third root is also real ! The reason : because complex roots come by pairs : if you have a complex root, the complex conjugate of this root is also a root. – Jean Marie Jul 24 '23 at 13:16
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@JeanMarie - but two of three real roots could be the same. – Paul Sinclair Jul 25 '23 at 17:22
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@Paul Sinclair You are right... – Jean Marie Jul 25 '23 at 18:10
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This discriminant is in fact classical once you have factored out $a$ : see here – Jean Marie Jul 26 '23 at 07:53