The Quadratic Formula - Khan Academy Lecture

Introduction

• The quadratic formula is one of the most useful formulas in mathematics.
• It helps solve quadratic equations by finding the roots or zeros.
• The formula is: [ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]
• It's recommended to memorize the formula and understand its derivation.

Components of a Quadratic Equation

• Standard form: ( ax^2 + bx + c = 0 )
  • a: Coefficient of ( x^2 )
  • b: Coefficient of ( x )
  • c: Constant term

Solving Using the Quadratic Formula

1. Identify coefficients:
   • Example: For ( x^2 + 4x - 21 = 0 ), ( a = 1 ), ( b = 4 ), ( c = -21 )
2. Apply the formula:
   • Substitute values into the formula.
   • Perform calculations step-by-step.
3. Example calculation:
   • ( x = \frac{-4 \pm \sqrt{16 + 84}}{2} = \frac{-4 \pm 10}{2} )
   • Solutions: ( x = 3 ) or ( x = -7 )

Importance of the Quadratic Formula

• Works even when factoring is difficult or impossible.
• Indicates when there are no real solutions (when the discriminant ( b^2 - 4ac ) is negative).

Example of No Real Solutions

• Equation: ( 3x^2 + 6x + 10 = 0 )
• Discriminant calculation:
  • ( b^2 - 4ac = 36 - 120 = -84 )
  • Negative discriminant implies no real solutions.
• Visualization using graphing shows no x-axis intersections.

Example with Imaginary Solutions

• Equation: ( -3x^2 + 12x + 1 = 0 )
• Calculation:
  • ( x = \frac{-12 \pm \sqrt{156}}{-6} )
  • Factor the square root of ( 156 ) to simplify.
  • Results in ( 2 \pm \frac{\sqrt{39}}{3} )
• Graph confirms solutions approximately around x = 4 and x < 1.

Conclusion

• The quadratic formula provides a reliable method to solve quadratic equations.
• Understanding its use and implications, such as when solutions are not real, is crucial.
• Practice using the formula will enhance problem-solving skills in algebra.

Additional Resources

• Further examples and explanations can be found on Khan Academy's platform.