Quadratic Equations

A quadratic equation is a second-degree polynomial equation in a single variable x.

The general form of a quadratic equation is:

ax² + bx + c = 0

Where x represents the variable, and a, b, and c are constants with a ≠ 0.

Ways to Solve Quadratic Equations

1. Factoring: If the quadratic equation can be factored, setting each factor equal to zero will give the solutions.
2. Quadratic Formula: The quadratic formula can be used to find the solutions for any quadratic equation.
3. Completing the Square: This method involves manipulating the equation to create a perfect square trinomial, from which the solutions can be found.
4. Graphing: Graphing the equation and finding the x-intercepts can give the solutions.

Important Concepts

• Discriminant: The expression b² - 4ac, which determines the nature of the solutions.
  • If the discriminant is positive, the equation has two distinct real solutions.
  • If the discriminant is zero, the equation has one real solution (a repeated root).
  • If the discriminant is negative, the equation has no real solutions (two complex conjugate solutions).
• Vertex: The vertex of the parabola represented by the quadratic equation is located at the point (h, k), where h = -b/2a and k = f(h).
• Axis of Symmetry: The line x = -b/2a, which passes through the vertex of the parabola and divides it into two symmetric halves.

Practice Problems

Solve the following quadratic equations using the method of your choice:

1. 2x² - 5x + 2 = 0
2. x² + 4x + 4 = 0
3. 3x² - 7x - 6 = 0

Applications

Quadratic equations are widely used in physics, engineering, economics, and many other fields to model various real-world phenomena such as motion, trajectories, optimization, and more.