Discriminant of cubic equation in terms of coefficients

In summary: The symmetry of the coefficients means that exchanging the roots in the expressions (x1 and x3 for instance) does not change the coefficients of the remaining terms. So if you want to find the coefficients of a term in the expression, just multiply the coefficient of that term by the first equation and then the coefficient of the term in the second equation, and you'll have the coefficients of the term in the expression.In summary, the author is trying to find the coefficients of a term in the expression (x1 - x2)2(x2 - x3)2(x3 - x1)2. He suggests that there may be such an expression, that it is symmetrical, and that the coefficients of
  • #36
Cool, and thanks for your kind words epenguin. :smile: And thank you so much for your help, Dick and epenguin, I am grateful to you for you took the time and effort to help me with this!
 
<h2> What is the discriminant of a cubic equation?</h2><p>The discriminant of a cubic equation is a mathematical term that is used to determine the nature of the roots of the equation. It is calculated using the coefficients of the equation and can help determine if the equation has real or imaginary roots.</p><h2> How is the discriminant of a cubic equation calculated?</h2><p>The discriminant of a cubic equation can be calculated using the formula D = b^2c^2 - 4ac^3 - 4b^3d - 27a^2d^2 + 18abcd, where a, b, c, and d are the coefficients of the cubic equation ax^3 + bx^2 + cx + d = 0.</p><h2> What does a positive discriminant indicate?</h2><p>A positive discriminant indicates that the cubic equation has three distinct real roots. This means that the equation crosses the x-axis three times and has three solutions.</p><h2> What does a negative discriminant indicate?</h2><p>A negative discriminant indicates that the cubic equation has one real root and two complex roots. This means that the equation does not cross the x-axis and has only one real solution.</p><h2> What does a zero discriminant indicate?</h2><p>A zero discriminant indicates that the cubic equation has at least two equal real roots. This means that the equation touches the x-axis at two points and has two equal solutions.</p>

FAQ: Discriminant of cubic equation in terms of coefficients

What is the discriminant of a cubic equation?

The discriminant of a cubic equation is a mathematical term that is used to determine the nature of the roots of the equation. It is calculated using the coefficients of the equation and can help determine if the equation has real or imaginary roots.

How is the discriminant of a cubic equation calculated?

The discriminant of a cubic equation can be calculated using the formula D = b^2c^2 - 4ac^3 - 4b^3d - 27a^2d^2 + 18abcd, where a, b, c, and d are the coefficients of the cubic equation ax^3 + bx^2 + cx + d = 0.

What does a positive discriminant indicate?

A positive discriminant indicates that the cubic equation has three distinct real roots. This means that the equation crosses the x-axis three times and has three solutions.

What does a negative discriminant indicate?

A negative discriminant indicates that the cubic equation has one real root and two complex roots. This means that the equation does not cross the x-axis and has only one real solution.

What does a zero discriminant indicate?

A zero discriminant indicates that the cubic equation has at least two equal real roots. This means that the equation touches the x-axis at two points and has two equal solutions.

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