nature of the roots and discriminant
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This document provides guidance on identifying the nature of roots of quadratic equations. It begins by identifying the least mastered skill of identifying the nature of roots. It then reviews the standard form of a quadratic equation and the quadratic formula. The key concept of the discriminant is explained, which determines the number and type of roots. Examples are provided to show how to find the discriminant and use it to describe the nature of roots. Activities are included for students to practice rewriting equations in standard form, finding a, b, c values, calculating discriminants, and determining the nature of roots. An assessment card with example problems is provided to check understanding.
nature of the roots and discriminant
- 1. Prepared by: Maricel T. Mas Lipay High School Strategic Intervention Material in Mathematics-IX The Nature of the Roots and The Discriminant
- 2. Guide Card Least Mastered Skill: • Identify the Nature of the Roots Sub tasks: Identify values of a, b and c of a quadratic equation, Find the discriminant; and Describe the nature of roots of quadratic equation.
- 3. The Standard Form of Quadratic Equation is… ax2 + bx + c = 0 The Quadratic Formula is… 2 4 2 b b ac x a
- 4. 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.
- 5. WHY IS THE DISCRIMINANT IMPORTANT? The discriminant tells you the number and types of answers (roots) you will get. The discriminant can be +, –, or 0 which actually tells you a lot! Since the discriminant is under a radical, think about what it means if you have a positive or negative number or 0 under the radical. ???
- 6. How to find the discriminant? Example 1: Find the discriminant of x 2 – 2x – 15 = 0 Step 2: Identify the value of a, b and c a = 1 b = -2 c = -15 Step 3: Substitute these values to b 2 – 4ac Step 1: Write first the equation into standard form Solution: D = b 2 – 4ac D = (-2) 2 – 4(1)(15) D = 64
- 7. Activity No. 1.a : Set Me To Your Standard Now it’s your turn Directions: Rewrite each quadratic equation in standard form. 1 x 2 – 5x = 14 2. 2x 2 + x = 5 3. x 2 + 25 = 10x 4. 4x 2 = 9x - 7 5. 3x 2 + 2x = 5
- 8. Activity No. 1.b Now it’s your turn Directions: Using the given quadratic equations on activity no 1.b, identify the values of a, b, and c. 1. x 2 – 5x – 14 = 0 2. 2x 2 + x = 5 3. x 2 + 25 = 10x 4. 4x 2 – 9x + 7 = 0 5. 3x 2 + 2x - 5 = 0 a = ___ b = ___ c = ___ a = ___ b = ___ c = ___ a = ___ b = ___ c = ___ a = ___ b = ___ c = ___ a = ___ b = ___ c = ___
- 9. Activity No. 2 Directions: Using the values of a, b, and c of Activity No. 1, find the discriminant of the following using b 2 – 4ac: 1. x 2 – 5x – 14 = 0 2. 2x 2 + x = 5 3. x 2 + 25 = 10x 4. 4x 2 – 9x + 7 = 0 5. 3x 2 + 2x - 5 = 0 a. 81 b. 11 c. -31 a. 39 b. - 39 c. 41 a. 0 b. 1 c. 100 a. - 31 b. 31 c. 81 a. -56 b. -64 c. 64
- 10. Let’s evaluate the following equations. 1. x2 – 5x – 14 = 0 What number is under the radical when simplified? D=81 b2 – 4ac > 0, perfect square The nature of the roots : REAL, RATIONAL, UNEQUAL 2. ) 2x2 + x – 5 = 0 What number is under the radical when simplified? D= 41 b2 – 4ac > 0, not a perfect square The nature of the roots: REAL, IRRATIONAL, UNEQUAL 4.) 4x2 – 9x + 7 = 0 What number is under the radical when simplified? D = –31 b2 – 4ac < 0, (negative) The nature of the roots: imaginary 3.) x2 – 10x + 25 = 0 What number is under the radical when simplified? D = 0 b2 – 4ac = 0 The nature of the roots: REAL, RATIONAL, EQUAL
- 11. Determine whether the given discriminant is a)greater than zero, perfect square b) Greater than zero, not a perfect square c) Equals zero d) Less than zero ____1) 95 ____2) 225 ____3) -9 ____4) 0 ____5) 63 Activity No. 3
- 12. Activity # 4 Determine whether the given discriminant is a) real, rational, equal b) real, rational, unequal c) real, irrational, unequal d) imaginary ____1) 12 ____2) 0 ____3) 49 ____4) -5 ____1) 27
- 13. Activity No. 5: Try These. For each of the following quadratic equations, a) Find the value of the discriminant, and b) Describe the number and type of roots. ____1) x 2 + 14x + 49 = 0 ____2) . x 2 + 5x – 2 = 0 ____3) 3x 2 + 8x + 11 = 0 ____4) x 2 + 5x – 24 = 0 D=____, ____________________ D=____, ____________________D=____, ____________________ D=____, ____________________
- 14. Assessment Card No. 1: Write the values of a, b & c in the quadratic equation, then check the discriminant and nature of roots of quadratic equation . 1. x2 – 8x + 15 = 0 I. a = ___ b = ___ c = ___ II. __ 4 __) 0 __ ) -4 __real, rational, equal __real, rational, unequal __real, irrational, unequal __imaginary 2. 2x2 + 4x + 4 = 0 I. a = ___ b = ___ c = ___ II. __) 16 __) 0 __ ) -16 __real, rational, equal __real, rational, unequal __real, irrational, unequal __imaginary
- 15. 3. 3x2 + 12x + 12 = 0 I. a = ___ b = ___ c = ___ II. __) 4 __) 0 __ ) -4 __real, rational, equal __real, rational, unequal __real, irrational, unequal __imaginary 4. 8x2 - 9x + 11 = 0 I. a = ___ b = ___ c = ___ II. __) -172 __) -721 __ ) -271 __real, rational, equal __real, rational, unequal __real, irrational, unequal __imaginary
- 16. Enrichment: Directions: Determine the nature of the roots of the following quadratic equations.
- 17. Answer Card Activity No. 1.a 1. x2 – 5x – 14 =0 2. 2x2 + x – 5 = 0 3. x2 -10x + 25 = 0 4. 4x2 – 9x + 7 = 0 5. 3x2 + 2x – 5 = 0 Activity No. 1.b. 1. a = 1 b = -5 c=-14 2. a = 2 b = 1 c = -5 3. a = 1 b = -10 c = 25 4. a = 4 b = -9 c = 7 5. a = 3 b = 2 c = -5 Activity No. 2 1. a. 81 2. c. 41 3. a. 0 4. a. -31 5. c. 64 Activity No. 3 1. b 2. a 3. d 4. c 5. b Activity No. 4 1. c 2. a 3. b 4. d 5. b Activity No. 5 1. D=0,real, rational, equal 2. D= 33, real, irrational, unequal 3. D= -68, imaginary 4. D= 121, real, rational, unequal 1.) I. a=1 b= -8 c=15 II. 4 III. real, rational, unequal 2.) I. a= 2 b = 4 c = 4 II. -16 III. imaginary 3.) I. a= 3 b= 12 c= 12 II. 0 III. real, rational, unequal 4.) I. a= 8 b= -9 c= 11 II. -271 III. imaginary
- 18. References Jose-Dilao, Soledad, Orines, and Bernabe, Julieta G. Advanced Algebra, Trigonometry and Statistics IV, SD Publications, Inc, 2009, p. 73 Learner’s Material Mathematics – Grade 9 First Edition, 2014 pp. 65-70.