Quadratics

1. Quadratic Equations

• Definition: A quadratic equation is an equation of the form:

  • ax² + bx + c = 0

  • where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero.

• Solutions (Roots): The values of 'x' that satisfy the equation. A quadratic equation can have:

  • Two distinct real solutions

  • One repeated real solution (also called a double root)

  • No real solutions (complex roots)

2. Solving Quadratic Equations

• Factoring:

  • Express the quadratic equation as a product of two linear factors.

  • Example: x² - 5x + 6 = (x - 2)(x - 3) = 0

• Quadratic Formula:

  • A general formula to find the solutions of any quadratic equation:

    • x = (-b ± √(b² - 4ac)) / 2a

• Completing the Square:

  • A method for rewriting the quadratic equation in vertex form.

3. Quadratic Functions

• Definition: A function of the form:

  • f(x) = ax² + bx + c

  • where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero.

• Graph: The graph of a quadratic function is a parabola, which is a U-shaped curve.

• Key Features of a Parabola:

  • Vertex: The highest or lowest point on the parabola.

  • Axis of Symmetry: A vertical line that divides the parabola into two symmetrical halves.

  • x-intercepts: The points where the parabola intersects the x-axis (also the roots of the quadratic equation).

  • y-intercept: The point where the parabola intersects the y-axis.

4. Applications of Quadratics

• Physics: Projectile motion, the path of a thrown object.

• Engineering: Designing structures, optimizing shapes.

• Economics: Maximizing profit, minimizing costs.

• Mathematics: Many other areas of mathematics build upon the foundation of quadratic equations and functions.