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• 1. YES-NO-STAND-UP Game Questions: 1. Do you like chocolates? 2. Do you like to hurt people? 3. Do you love math? 4. Is 2 + 2 = 5? 5. Is 9= 3 ? 6. Is 4 a perfect square? 7. Is 1 x 2 = 3? 8. Is 100 not a perfect square? 9. Is there a way to predict the nature of the root? 10. Are you familiar with discriminant in a quadratic? equation?
• 2. NATURE OF ROOTS OF QUADRATIC EQUATION
• 3. At the end of this lesson, the students are expected to: Objectives b. describes the nature of roots of a quadratic equation through its discriminant; c. value the importance of discriminant through the roots of the quadratic equation. a. identifies the nature of roots of a given quadratic equation;
• 4. Group Activity 1. 𝒙𝟐 + 𝟒𝒙 + 𝟑 = 𝟎 2. 𝒙𝟐 − 𝟏 = 𝟎 Boys vs. Girls Direction: Find the roots of the quadratic eaquation using quadratic formula. Boys group will answer #1 and girls group will answer the #2. The groups have 5 minutes to answer and choose a representative to present their answer.
• 5. Questions from the activity: 1. What is the standard form of the quadratic equation? 2. What is the formula for quadratic formula? 3. What is the formula of a discriminant? 4. Why do we need to determine the discriminant of the quadratic equation?
• 6.
• 7. WHY USE THE QUADRATIC FORMULA?  The quadratic formula allows you to solve ANY quadratic equation, even if you cannot factor it.  An important piece of the quadratic formula is what’s under the radical: b2 – 4ac  This piece is called the discriminant.
• 8. WHY IS THE DISCRIMINANT IMPORTANT? The value of the expression 𝒃𝟐 – 4ac is called the discriminant of the quadratic equation ax² + bx + c = 0. This value can be used to describe the nature of the roots of a quadratic equation. It can be zero, positive and perfect square, positive but not perfect square or negative. ???
• 9. Nature of Roots of Quadratic Equation Discriminant • It is the number being used to describe the nature of roots of a quadratic equation. Formula: d = b2 – 4ac provided that the equation is in standard form. discriminant Nature of Roots d = 0 real numbers and equal roots d > 0 d is a perfect square rational numbers and unequal roots d is not a perfect square irrational numbers and unequal roots d < 0 No real roots
• 10. 9 Remember: Equation must be in standard form Example 1: Describe the nature of roots of 3t2 + 5t = 2 Solution: 3t2 + 5t = 2 Step 1: Write first the equation into standard form: 3t2 + 5t – 2 = 0 Step 2: Identify a, b and c a = 3, b = 5, c = –2 Step 3: Substitute these values to d = b2 – 4ac d = (5)2 – 4(3)(–2) d = 25 + 24 d = 49 Nature of Roots: rational numbers and equal discriminant Nature of Roots d = 0 real numbers and equal roots d > 0 d is a perfect square rational and unequal roots d is not a perfect square irrational and unequal roots d < 0 No real root
• 11. Remember: Equation must be in standard form Example 2: Compute for the discriminant of (x – 2)2 = 0 Solution: Step 1: Write first the equation into standard form: x2 – 4x + 4 = 0 Step 2: Identify a, b and c a = 1, b = –4, c = 4 Step 3: Substitute these values to d = b2 – 4ac d = (–4)2 – 4(1)(4) d = 16 – 16 d = 0 Nature of Roots: real numbers and equal discriminant Nature of Roots d = 0 real numbers and equal d > 0 d is a perfect square rational numbers and unequal d is not a perfect square irrational numbers and unequal d < 0 No real roots
• 12. Remember: Equation must be in standard form Example 3: What is the nature of roots of h2 – 16 = – 3h Solution: Step 1: Write first the equation into standard form: h2 + 3h – 16 = 0 Step 2: Identify a, b and c a = 1, b = 3, c = –16 Step 3: Substitute these values to d = b2 – 4ac d = (3)2 – 4(1)(–16) d = 9 + 64 d = 73 Nature of Roots: irrational numbers and unequal discriminant Nature of Roots d = 0 real numbers and equal d > 0 d is a perfect square rational numbers and unequal d is not a perfect square irrational numbers and unequal d < 0 No real roots
• 13. Remember: Equation must be in standard form Example 4: Describe the nature of roots of the equation k2 - 3k = - 6 Solution: Step 1: Write first the equation into standard form: k2 - 3k + 6 = 0 Step 2: Identify a, b and c a = 1, b = -3, c = 6 Step 3: Substitute these values to d = b2 – 4ac d = (-3)2 – 4(1)(6) d = 9 – 24 d = – 15 Nature of Roots: No real roots discriminant Nature of Roots d > 0 d is a perfect square • Two real, rational and unequal roots d is not a perfect square • Two real, irrational and unequal roots d = 0 • One real and rational root. d < 0 • No real root discriminant Nature of Roots d = 0 real numbers and equal d > 0 d is a perfect square rational numbers and unequal roots d is not a perfect square irrational numbers and unequal roots d < 0 No real root
• 14. Remember: Equation must be in standard form Example 5: Determine the nature of roots of the equation 3x2 + 11x + 8 = 0 Solution: Step 1: Write first the equation into standard form: 3x2 + 11x + 8 = 0 Step 2: Identify a, b and c a = 3, b = 11, c = 8 Step 3: Substitute these values to d = b2 – 4ac d = (11)2 – 4(3)(8) d = 121 – 96 d = 25 Nature of Roots: rational numbers and unequal discriminant Nature of Roots d = 0 real numbers and equal d > 0 d is a perfect square rational numbers and unequal roots d is not a perfect square irrational and unequal roots d < 0 No real root
• 15. Activity Place Me on the Table! Direction: Answer the following. 1. Complete the table below. Equation b²-4ac Nature of roots 1. x² + 5x + 4 = 0 2. 2x² + x - 21= 0 3. 4x² +4x + 1 = 0 4. x² - 2x - 2 = 0 5. 9x² + 16= 0 2. How would you describe the roots of quadratic equation when the value of b²- 4ac is 0? Positive and Perfect Square? Positive but not Perfect Square? Negative? . 3. Which quadratic equation has roots that are real numbers and equal? Rational numbers? Irrational numbers? Not real numbers? 4. How do you determine the quadratic equation having roots that are real numbers and equal? Rational numbers? Irrational numbers? Not real numbers?
• 16. Directions: Determine the discriminants and nature of the roots of the following quadratic equations. Write your answer on the space provided. Evaluation 2. 𝑥2 + 9𝑥 + 20 = 0 1. 𝑥2 + 6𝑥 + 9 = 0 3. 2𝑥2 − 10𝑥 + 8 = 0 4. 𝑥2 + 5𝑥 + 10 = 0 5. 𝑥2 + 6𝑥 + 3 = 0 discriminant: discriminant: discriminant: discriminant: discriminant: nature of the roots nature of the roots nature of the roots nature of the roots nature of the roots
• 17. Assignment: Study Sum and Product of Roots of a Quadratic Equation.
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