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July 23rd, 2015, 10:52 AM   #1
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Conditions for different roots of cubic equation.

I realized the discriminant for cubic can be 123.JPG

but I don't know what conditions are needed for different results of roots.

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July 23rd, 2015, 01:56 PM   #2
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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.
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July 23rd, 2015, 02:25 PM   #3
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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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