Mathematics 1 - Quadratic Equations

Definition:

• A quadratic equation is a second-degree polynomial equation in one variable. It has the form: ax^2 + bx + c = 0
• Here, a, b, and c are constants, and x is the variable. a cannot be equal to 0, otherwise, it wouldn't be a quadratic equation.

Solutions:

• Quadratic equations typically have two solutions, which can be real or complex numbers. These solutions are called roots or zeroes of the equation.
• The solutions can be found using methods like factoring, completing the square, quadratic formula, or graphical methods.

Discriminant:

• The discriminant (Δ) of a quadratic equation is given by: Δ = b^2 - 4ac
• The discriminant determines the nature of the roots:
  • If Δ > 0, the equation has two distinct real roots.
  • If Δ = 0, the equation has exactly one real root (the roots are repeated).
  • If Δ < 0, the equation has two complex roots.

Vertex:

• The vertex of a quadratic function in the form y = ax^2 + bx + c is given by the point (h, k), where: h = -b/2a and k = f(h) = ah^2 + bh + c

Graph:

• The graph of a quadratic equation is a parabola. The direction of the parabola (upward or downward) depends on the sign of the leading coefficient a.
• The axis of symmetry of the parabola is a vertical line passing through the vertex.

Applications:

• Quadratic equations are widely used in various fields, including physics, engineering, economics, and computer science.
• They describe many natural phenomena, such as projectile motion, the shape of satellite dishes, and the pricing of products based on supply and demand.