Analyzing the coefficients of the quadratic equation

In summary, the conversation discussed the classification of the quadric equation and how it can be expressed in matrix form. The goal was to analyze the square matrix and determine if it is a straight line, hyperbola, circle, ellipse, parabola, etc. However, the attempt to express the equation in this way did not work and help was requested.
  • #1
Bruno Tolentino
97
0
Is possible classify the quadric equation Axx + Bxy + Cyx + Dyy + Ex + Fy + G = 0 how straight, hyperbola, circle, ellipse, parabola, etc, in the same way that is did in the phase plan:

https://upload.wikimedia.org/wikipedia/commons/3/35/Phase_plane_nodes.svg

https://upload.wikimedia.org/wikipedia/commons/3/35/Phase_plane_nodes.svg

I tried this, but, for some reason, don't works... Can you help me!?
 
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  • #3
I want to express the quadric equation in this way: [tex]
\begin{bmatrix}
A & B \\
C & D
\end{bmatrix}
:
\begin{bmatrix}
x^2 & xy \\
yx & y^2
\end{bmatrix}
+
\begin{bmatrix}
E\\
F
\end{bmatrix}
\cdot
\begin{bmatrix}
x\\
y
\end{bmatrix}
+
G=0
[/tex]

And I want to analyze the square matrix, the first matrix, the matrix of the coefficients.
 

FAQ: Analyzing the coefficients of the quadratic equation

1. What is the quadratic equation?

The quadratic equation is a mathematical formula used to solve quadratic functions, which are equations of the form ax^2+bx+c=0. It is written as x = (-b ± √(b^2-4ac))/2a.

2. What are the coefficients in the quadratic equation?

The coefficients in the quadratic equation are the numbers that appear in front of the variable terms. In the standard form of the quadratic equation, ax^2+bx+c=0, a, b, and c are the coefficients.

3. How do you analyze the coefficients in the quadratic equation?

The coefficient a determines the shape and direction of the parabola, with positive values creating a "U" shape and negative values creating an upside-down "U" shape. The coefficient b affects the location of the parabola on the x-axis, while c determines the y-intercept. By analyzing these coefficients, you can determine the characteristics of the quadratic function, such as the vertex, x-intercepts, and direction of opening.

4. What do the coefficients tell us about the solutions of the quadratic equation?

The coefficients can tell us whether the quadratic equation has two distinct real solutions, one repeated real solution, or no real solutions. If the discriminant (b^2-4ac) is positive, there are two distinct real solutions. If it is zero, there is one repeated real solution. If it is negative, there are no real solutions.

5. How can analyzing the coefficients help in solving quadratic equations?

By analyzing the coefficients, you can determine the type of solutions the quadratic equation will have, which can guide you in choosing the appropriate method for solving. If there are two distinct real solutions, the quadratic formula can be used. If there is one repeated real solution, factoring can be used. If there are no real solutions, the equation can be solved using complex numbers.

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