Summary of Second Degree Equation: Coefficients

KEYWORDS

What defines a second degree equation?
What are the coefficients of a second degree equation and what is their importance?
How does the discriminant (Δ) influence the number and nature of the roots?
What is the relationship between the coefficients and the roots of the equation?

Recognition of the general form of the second degree equation: ax² + bx + c = 0
Identification and interpretation of the coefficients a, b, and c
Calculation of the discriminant (Δ) and its application in Bhaskara's Theorem
Application of Girard's relations: sum (x₁ + x₂ = -b/a) and product (x₁ * x₂ = c/a) of the roots

General form of the second degree equation: ( ax^2 + bx + c = 0 )
Discriminant (Delta - Δ): ( Δ = b^2 - 4ac )
Bhaskara's formula (roots of the equation): ( x_{1,2} = \frac{-b \pm \sqrt{Δ}}{2a} )
Girard's relations (sum and product):
- Sum of the roots: ( x₁ + x₂ = -\frac{b}{a} )
- Product of the roots: ( x₁ * x₂ = \frac{c}{a} )

Second degree equation: An algebraic expression that has its highest power in the quadratic term (ax²).
Coefficients (a, b, c):
- a: Leading coefficient or quadratic coefficient; multiplies the degree two term and cannot be zero.
- b: Linear coefficient; multiplies the degree one term.
- c: Constant term; does not multiply any variable.

Solving second degree equations is fundamental for various areas of mathematics and applied sciences.
The value of the discriminant (Δ) determines the number and type of roots of the equation (real and distinct, real and equal, or complex).

General Form of the Equation: Knowing that every second degree equation can be rewritten as ax² + bx + c = 0.
Discriminant (Δ): Understanding that Δ = b² - 4ac provides information about the roots. If Δ > 0, two real and distinct roots; if Δ = 0, a real double root; if Δ < 0, complex roots.
Bhaskara's Theorem: Understanding that the formula ( x_{1,2} = \frac{-b \pm \sqrt{Δ}}{2a} ) allows finding the roots of the equation from the coefficients and the discriminant.
Girard's Relations: Recognizing that the relations between coefficients and roots allow expressing the sum and product of the roots in terms of the coefficients: ( x₁ + x₂ = -\frac{b}{a} ) and ( x₁ * x₂ = \frac{c}{a} ).

Coefficient a: Affects the concavity of the parabola represented by the equation in the Cartesian plane.
Coefficient b: Influences the position of the symmetry axis of the parabola and, consequently, the roots on the graph.
Term c: Represents the point where the parabola intercepts the y-axis on the graph.

Example of Discriminant (Δ):
- Consider the equation 2x² - 4x + 2 = 0.
- Calculation of Δ: Δ = (-4)² - 4(2)(2) = 16 - 16 = 0.
- Conclusion: The equation has a real double root, found using Bhaskara's formula.
Case of Girard's Relations:
- For the equation x² - 5x + 6 = 0, we identify a = 1, b = -5, and c = 6.
- Applying the relations, we have:
  - Sum of the roots: x₁ + x₂ = -(-5)/1 = 5.
  - Product of the roots: x₁ * x₂ = 6/1 = 6.
- Verification: The roots found are 2 and 3, whose sum is 5 and product is 6, according to Girard's relations.

Second Degree Equation: An equation characterized by the presence of a quadratic term, ax², and described in the form ax² + bx + c = 0, where a ≠ 0.
Coefficients (a, b, c): The value of 'a' determines the opening of the parabola, 'b' affects the horizontal displacement and 'c' is the intercept on the y-axis.
Discriminant (Δ): Defines the number and type of roots (real distinct, real double or complex) and is calculated by b² - 4ac.
Girard's Relations: Relate the coefficients to the values of the roots through the sum (-b/a) and product (c/a).

Understanding the coefficients is fundamental to understanding the structure and solution of the second degree equation.
The discriminant (Δ) is a key piece for identifying the nature of the roots without necessarily calculating the roots.
Girard's Relations allow anticipating the sum and product of the roots, bringing a deeper insight into the properties of the equation.
The ability to manipulate and interpret these elements is essential for solving mathematical problems and applying these concepts in practical contexts.