MODULE 3

• Quadratic equation


  An equation of the form ax^2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0
• Roots of a quadratic equation

  • Can be real, rational, irrational, or imaginary
  • Determined by the discriminant b^2 - 4ac
• Determining the nature of the roots of a quadratic equation

  1. Find the discriminant b^2 - 4ac

  2. If discriminant > 0, then two real roots

  3. If discriminant = 0, then one real root

  4. If discriminant < 0, then two imaginary roots
• Discriminant


  The expression b^2 - 4ac, which determines the nature of the roots of a quadratic equation
• If discriminant is positive


  Equation has two distinct real roots
• If discriminant is a perfect square


  Roots are real, rational, and unequal
• If discriminant is positive but not a perfect square


  Roots are real, irrational, and unequal
• If discriminant is zero


  Equation has one real root
• If discriminant is negative


  Equation has no real roots, only imaginary roots
• Sum of the roots

  • b/a
• Product of the roots


  c/a
• Finding the quadratic equation given the sum and product of the roots

  1. Let S = sum of roots and P = product of roots

  2. The quadratic equation is x^2 - Sx + P = 0
• Rodolfo wants to make rectangular plots in his backyard where the length is 2 meters longer than the width, and the area is 10 m^2
• Aling Nena wants to make a table where the length is 1 meter longer than the width, and the area is 6 m^2
• Determining the nature of the roots of Quadratic Equation

  1. Write a journal on your own understanding

  2. Provide at least 5 own examples of quadratic equations

  3. Determine the nature of roots of quadratic equations
• Aling Nena's table


  Length has to be one (1) meter longer than its width<|>Area of the table is (x)(x + 1) = 6 m2
• a, b, c


  Values to be found
• Discriminant


  Value to be determined
• Nature of roots

  To be described
• Scoring Rubrics
• Criteria

  • Neatness
  • Accuracy
  • Creativity
  • Time Management
• Personal insights and observations in doing this performance task
• Personal realizations on how this task helps you see the real-world application of the topic: Nature of the roots of Quadratic Equations
• Discriminant


  The value b2 - 4ac
• The sum of the roots of a quadratic equation is equal to the negation of the coefficient of the second term, divided by the leading coefficient
• The product of the roots of a quadratic equation is equal to the constant term (the third term), divided by the leading coefficient
• Completing the table: Sum and product of roots


  • x2 - x + 1 = 0
  • x2 + x - 1 = 0
• Finding quadratic equations given sum and product of roots

  1. Sum = -2, product = 5

  2. Sum = 3, product = -8

  3. Sum = -1/3, product = -1/3
• Key concepts

  • Sum of roots
  • Product of roots
  • Standard form of quadratic equation
  • Substituting
  • Multiplying
  • Dividing
  • LCD
• Is 7x + 3 = 0 equal to 7(x + 3)2 = 0? Will the two equations yield the same sum and product of the roots? Explain why? Explain why not?
• Make a simple scrapbook


  1. Use bond paper and colored paper

  2. Include all the things you have learned in this lesson
• Journal on how to determine a quadratic equation

  1. Given the roots

  2. Given the sum and products of roots
• Finding the quadratic equations

  • Given the roots
  • Given the sum and product of the roots