Discriminant of a quadratic equation in 2 variables

In summary, the discriminant of a general equation of 2 degree in 2 variables is typically defined as abc+2gfh-a(f sq.)-b(g sq.)-c(h sq.). However, there are variations in its formula, such as the one provided by Wolfram for quadratic curves. Its significance and implication may vary depending on the context and application.
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Can anyone tell me how to calculate the discriminant of a general equation of 2 degree in 2 variables, ax[itex]^{2}[/itex]+by[itex]^{2}[/itex]+2gx+2fy+2hxy+c=0? Thanks!
 
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  • #2
I've never seen a textbook that defined the descriminant for a polynomial of two variables. How do your materials define it?
 
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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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The discriminant of a quadratic equation in 2 variables is a mathematical term that represents the nature of the roots of the equation. In the general equation of 2 degrees in 2 variables, ax^{2}+by^{2}+2gx+2fy+2hxy+c=0, the discriminant can be calculated using the following formula: b^{2}-4ac-4h^{2}. This formula involves the coefficients of the equation, a, b, c, g, f, and h.

To calculate the discriminant, substitute the values of these coefficients into the formula and simplify the expression. The resulting value will determine the nature of the roots of the equation. If the discriminant is positive, the equation will have two distinct real roots. If the discriminant is zero, the equation will have one real root. And if the discriminant is negative, the equation will have two complex roots.

The discriminant is an important factor in understanding the behavior of quadratic equations in 2 variables. It helps determine the number and type of solutions the equation will have, which can have significant implications in various scientific and mathematical applications.
 

1. What is the discriminant of a quadratic equation in 2 variables?

The discriminant of a quadratic equation in 2 variables is a mathematical term that helps determine the nature of the solutions of the equation. It is represented by the symbol "Δ" and is calculated as b²-4ac, where a, b, and c are the coefficients of the equation.

2. How is the discriminant used to determine the nature of the solutions?

The value of the discriminant can be used to classify the solutions of a quadratic equation in 2 variables into three cases:
1) If Δ > 0, the equation has two distinct real solutions.
2) If Δ = 0, the equation has one real solution.
3) If Δ < 0, the equation has two complex solutions.

3. What does a positive discriminant indicate?

A positive discriminant (Δ > 0) indicates that the quadratic equation has two distinct real solutions. This means that the equation intersects the x-axis at two different points, giving two solutions for the variables x and y.

4. Can the discriminant be negative?

Yes, the discriminant can be negative (Δ < 0). This indicates that the quadratic equation has two complex solutions, which cannot be represented on a traditional x-y plane. Instead, they are represented in the complex plane as imaginary numbers.

5. How is the discriminant related to the graph of a quadratic equation in 2 variables?

The discriminant is related to the graph of a quadratic equation in 2 variables by helping to determine the number and nature of the solutions. A positive discriminant corresponds to a graph with two distinct x-intercepts, a zero discriminant corresponds to a graph with one x-intercept, and a negative discriminant corresponds to a graph with no x-intercepts (but instead has two complex solutions).

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