Quadratic Equations and Factored Form

This page provides a comprehensive overview of quadratic equations and their various forms, focusing on the factored form and its applications in solving equations and analyzing graphs.

The factored form of a quadratic function is presented as f(x) = a(x-x₁)(x-x₂), where x₁ and x₂ are the roots of the equation. This form is crucial for understanding the behavior of the function.

Definition: The discriminant (Δ) of a quadratic equation ax²+bx+c=0 is given by the formula Δ = b²-4ac. It determines the nature and number of roots.

The page outlines three cases based on the discriminant:

1. When Δ > 0, the equation has two distinct real roots.
2. When Δ = 0, the equation has one repeated real root.
3. When Δ < 0, the equation has no real roots.

Example: For the equation x² - 6x + 1 = 0, the discriminant is calculated as Δ = (-6)² - 4(1)(1) = 36 - 4 = 32. Since Δ > 0, this equation has two distinct real roots.

The document also introduces the canonical form of a quadratic function: f(x) = a(x-x₀)², where x₀ represents the x-coordinate of the vertex of the parabola.

Highlight: The sign of the coefficient 'a' determines whether the parabola opens upward (a > 0) or downward (a < 0), which is crucial for understanding the function's behavior.

A step-by-step method for converting from standard form to canonical form is provided, using the example f(x) = 3x² + 3x - 1.

Vocabulary: The "coefficient dominant" refers to the leading coefficient 'a' in the quadratic function f(x) = ax²+bx+c.

The page concludes with information about the vertex form and how to find the coordinates of the vertex (α, β), where α = -b/(2a) and β is the y-coordinate of the vertex.

Quote: "Si a > 0, f a un minimum, si a < 0, f a un maximum" (If a > 0, f has a minimum; if a < 0, f has a maximum)

This comprehensive guide provides students with the tools to résoudre équation second degré en ligne and understand the graphical representation of quadratic functions.