Quadratic Equations

Quadratic Equations

Key Terms and Definitions

• Quadratic Equation: An equation in the form , where , , and are constants, and .
• Roots (Solutions): The values of that satisfy the quadratic equation.
• Discriminant: The part of the quadratic formula , used to determine the nature of the roots.
• Vertex: The highest or lowest point on the graph of a quadratic function, located at .

Worked Example: Solve

1. Step 1: Factorize the quadratic equation:
   Find two numbers that multiply to and add to : and .
   Factorization:
2. Step 2: Solve for :
   Set each factor equal to :
   or
   Solutions: and
3. Verification: Substitute the solutions into the original equation:
   For :
   For :
   Both solutions are correct.

Tips and Tricks

• Always check if the quadratic equation can be factored before using other methods.
• If factoring is difficult, use the quadratic formula: .
• Understand the discriminant to predict the nature of the roots:
  • : Two distinct real roots
  • : One real root (repeated)
  • : No real roots (complex solutions)

Test-Taking Strategies

• Identify the method that best suits the quadratic equation (factoring, completing the square, or quadratic formula).
• Double-check your arithmetic when calculating the discriminant or solving for .
• Graph the equation when possible to confirm the number and location of roots.

Quadratic equations are fundamental in algebra and appear in various fields like physics, engineering, and economics. Mastering these equations prepares you for advanced topics such as polynomial functions, calculus, and optimization problems.

“The advancement and perfection of mathematics are intimately connected with the prosperity of the state.” – Napoleon Bonaparte