complex roots

Module 8: Quadratic Functions

Overview of Complex Roots

• Learning Outcomes:

  • Understand the introduction of complex numbers in the context of quadratic functions.

  • Find the complex roots of a quadratic function using the quadratic formula.

  • Use the discriminant to determine whether a quadratic function has real or complex roots.

Function Example

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  Consider the quadratic function:( f(x) = x^2 + 2x + 3 )

• Graphing the function shows it does not cross the x-axis (no x-intercepts).

Understanding Roots

• x-Intercepts:

  • 
    

    x-intercepts are found by setting the function equal to zero:( x^2 + 2x + 3 = 0 )

  • This function does not have real roots; it has complex roots instead.

Finding Complex Roots

• Quadratic Formula:

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    The quadratic formula is used for finding the roots:( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )

• For ( f(x) = x^2 + 2x + 3 ), applying the quadratic formula yields complex roots due to a negative discriminant.

The Discriminant

• Definition:

  • 
    

    The discriminant is the expression under the square root in the quadratic formula:( b^2 - 4ac )

• It indicates the nature of roots (real vs. complex):

  • If ( b^2 - 4ac > 0 ): Two distinct real solutions

  • If ( b^2 - 4ac = 0 ): One repeated real solution

  • If ( b^2 - 4ac < 0 ): Two complex solutions

Analyzing the Discriminant

• Discriminant values and their implications:

  • ( b^2 - 4ac = 0 ): One repeated rational solution

  • ( b^2 - 4ac > 0 ):

    • Perfect square: Two rational solutions

    • Not a perfect square: Two irrational solutions

  • ( b^2 - 4ac < 0 ): Two complex solutions

Practical Example of Discriminant

• Example Analysis:

  • For ( x^2 + 4x + 4 = 0 ): Discriminant is zero → one repeated root.

  • For ( 3x^2 - 5x - 2 = 0 ): Calculate discriminant to analyze roots.

Summary of Findings

• A quadratic equation can have:

  • Two real solutions

  • One real solution (repeated)

  • Two complex solutions

• Utilize the discriminant effectively to assess and find the nature of the solutions.