Quadratic Equations

Introduction

• Quadratic Equations are equations of degree 2, often expressed in the standard form:

  [ ax^2 + bx + c = 0 ]

• The term "Quadratic" comes from "quad" meaning square, as the variable x is squared (x^2).

• Examples:

  • [ 2x^2 + 5x + 3 = 0 ] with a=2, b=5, c=3
  • [ x^2 - 3x = 0 ] with a=1, b=-3, c=0
  • Not all equations with x are quadratic; e.g., [ 5x - 3 = 0 ] is not quadratic because a=0.

Solving Quadratic Equations

• Roots/Solutions: The solutions to a quadratic equation are where it equals zero, often called roots or zeros.
• Typically, there are two solutions.
• Methods to find solutions:

  • Factoring: Decomposing the equation into products.

  • Completing the Square

  • Quadratic Formula:

    [ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]

Quadratic Formula

• The "(\pm)" symbol means there are two solutions:
  • [ x = \frac{-b + \sqrt{b^2 - 4ac}}{2a} ]
  • [ x = \frac{-b - \sqrt{b^2 - 4ac}}{2a} ]
• Discriminant (b^2 - 4ac):
  • Positive: Two real solutions
  • Zero: One real solution (repeated)
  • Negative: Two complex solutions

Example

• Solve [ 5x^2 + 6x + 1 = 0 ]:
  • Coefficients: a=5, b=6, c=1
  • Using the Quadratic Formula:
    • [ x = \frac{-6 \pm \sqrt{36 - 20}}{10} ]
    • [ x = \frac{-6 \pm 4}{10} ]
    • Solutions: [ x = 0.2 ] or [ x = -1 ]

Complex Solutions

• Occur when the Discriminant is negative.
• Example: Solve [ 5x^2 + 2x + 1 = 0 ]:
  • Discriminant is negative: (b^2 - 4ac = -16)
  • Solutions include imaginary numbers: [ x = 0.2 \pm 0.4i ]

Memorizing the Formula

• Song to "Pop Goes the Weasel":
  • "x is equal to minus b, plus or minus the square root, of b-squared minus four a c, all over two a"

Summary

• Standard Form: [ ax^2 + bx + c = 0 ]
• Solutions depend on the Discriminant:
  • Positive: Two real roots
  • Zero: One real root
  • Negative: Two complex roots

• Tools: Quadratic Equation Solver, Factoring Quadratics, Completing the Square
• Additional Resources: Real World Examples, Graphing Quadratics, Derivation of the Equation