Discriminant of a quadratic equation in 2 variables

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SUMMARY

The discriminant of a quadratic equation in two variables, represented as ax² + by² + 2gx + 2fy + 2hxy + c = 0, is defined in the discussion as abc + 2gfh - a(f²) - b(g²) - c(h²). This formula is derived from the properties of conic sections and is crucial for determining the nature of the curve represented by the equation. The significance of the discriminant lies in its ability to classify the conic as an ellipse, hyperbola, or parabola based on its value. Additionally, Wolfram provides an alternative formula for the discriminant, which can be found at their dedicated page on quadratic curves.

PREREQUISITES
  • Understanding of quadratic equations and their general forms
  • Familiarity with conic sections and their classifications
  • Basic knowledge of algebraic manipulation and polynomial expressions
  • Access to mathematical resources such as Wolfram Alpha for further exploration
NEXT STEPS
  • Research the derivation of the discriminant for quadratic equations in two variables
  • Explore the classification of conic sections based on discriminant values
  • Study the implications of the discriminant in real-world applications, such as optimization problems
  • Examine alternative methods for calculating the discriminant using software tools like Mathematica or MATLAB
USEFUL FOR

Students, mathematicians, and educators interested in advanced algebra, particularly those studying conic sections and their properties.

Ashu2912
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Can anyone tell me how to calculate the discriminant of a general equation of 2 degree in 2 variables, ax^{2}+by^{2}+2gx+2fy+2hxy+c=0? Thanks!
 
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I've never seen a textbook that defined the descriminant for a polynomial of two variables. How do your materials define it?
 
In my text it is defined as abc+2gfh-a(f sq.)-b(g sq.)-c(h sq.). I just want to know how is it derived and what is it's significance and implication...
 

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