Department of Education MATHEMATICS 9 Illustrations of Quadratic Equations First Quarter – Week 1 EMALYN M. BALLONADO Writer DR. FELISA G. BASIJAN, CRISTINE CAROLINE C. GRATIS, CHARISMA JOY S. LULU, BENELIN G. RUMBAOA Validators DR. EMELITA D. BAUTISTA, ENGR. ROLANDO S. MULDONG, JOSEPH D. NILO, KRYSTELLE R. DUMLAO Quality Assurance Team Members Schools Division Office – Muntinlupa City Student Center for Life Skills Bldg., Centennial Ave., Brgy.Tunasan, Muntinlupa City (02) 8805-9935 / (02) 8805-9940 This module was designed to give you an opportunity to have a deeper understanding of quadratic equations. After going through this module, you are expected to: 1. illustrate quadratic equations. Directions: Choose the letter of the correct answer. 1. A mathematical sentence of degree 2 that can be written in the form ax2+bx+c = 0 where a, b, and c are real numbers and a is ≠ 0. A. Quadratic equation C. linear equation B. Quadratic inequality D. linear inequality 2. Which of the following is a quadratic equation? A. 2r2 + 4r - 1 C. x2 + 5x - 4 = 0 B. 3t - 7 = 2 D. 2a2 – 7a ≥ c = 0 3. In the quadratic equation 5x2 + 3x = 7, which is the quadratic term? A. x2 C. 5x2 B. -3x D. 4 4. What is the value of a in the quadratic equation 9x2 + 4x - 8 = 0? A. 4 B. 9 C. 0 D. -8 5. A quadratic equation of the form ax2+bx+c = 0 where a, b, and c are real numbers and a is ≠ 0 is said to be in ______________. A. slope form C. slope-intercept form B. Standard form D. normal form 6. What is the standard form of 3x + 5x2 = -7 A. 5x2 + 3x = -7 B. 3x + 5x2 + 7 = 0 C. 5x2 + 3x +7 = 0 D. 3x + 5x2 = -7 7. In the quadratic equation ax2 + bx + c = 0, a is ≠ ___. A. 1 B. 0 C. 2 8. Write the equation (2x + 7) (x - 1) = 0 in the standard form. A. 2x2 + 9x -7 = 0 C. 2x2 + 5x -7 = 0 B. 2x2 – 9x + 7 = 0 D. 2x2 – 5x + 7 = 0 2 D. −1 9. In the equation 5 - 2x2 = 6x, what is the value of a? A. 5 B. 6m C. -2 D. -5 10. Which of these equations describes a quadratic equation? A.2(x + 3)2 = 0 B. y + 3x2 - 0 C. 3x - 2 D. (x + 3) + 8 = 0 A. Find each indicated product: 1. 3(x2 + 7) 4. (x + 9) (x + 4) 2. 2x (x – 4) 5. (2t – 1) (t + 5) 3. 9(w + 7) (w + 3) 6. (8 – 3x) (8 + 3x) We start this module by accessing your knowledge of the different mathematics concepts previously studied and your skills in performing mathematical operations. These knowledge and skills will help you understand quadratic equations. As you go through this lesson, think of this important question: " how are quadratic equations used in solving real-life problems and in making decisions?" To find the answer, you have to read and understand first some important notes about quadratic equation. THE QUADRATIC EQUATION DEFINED If a linear equation is of the first degree, then the quadratic equation is of the second degree. And this is the simplest characteristic that distinguishes one from the other. A linear equation is in the form ax + b = 0, where a and b are real numbers and a ≠ 0. On the other hand, a quadratic equation takes the form ax² + bx + c = 0, where a, b, and c are all real numbers and a ≠ 0. These equations are both in standard form. 3 Illustrative Example Determine which of these are quadratic equations. 1. C = 2πr 3. x + 3x = 0 5. x² - 2x + 1 = 0 2. 2x² - 5x + 1 = 0 4. 3x² - 5 = 0 6. A = πr² Solution: Equations 2, 4, 5, and 6 are quadratic equations. A quadratic equation in one variable is a mathematical sentence of degree two that can be written in the following standard form, ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. In the equation, ax² is the quadratic term, bx is the linear term, and c is the constant term. Example 1: 3a² + 2a - 4 = 0 is a quadratic equation in the standard form with a = 3, b = 2 and c = -4. Example 2: 4x (x - 3) = 5 is a quadratic equation. However, it is not written in standard form. To write the equation in standard form, expand the product and make one side of the equation zero as shown below 4x (x - 3) = 5 4x²- 12x = 5 4x²- 12x - 5 = 5 - 5 4x² - 12x – 5 = 0 The equation becomes 4x² - 12x – 5 = 0, which is in the standard form. In the equation 4x² - 12x – 5 = 0, a = 4, b = -12 and c = -5 Example 3: The equation (2x + 5) (x - 1) = -6 is also a quadratic equation but it is not written in standard form same with example 2. The equation (2x + 5) (x - 1) = -6 can be written in standard form by expanding the product in making one side of the equation equal to zero as shown below (2x + 5) (x - 1) = -6 2x²- 2x + 5x -5 = -6 2x² + 3x -5 = -6 2x² + 3x -5 + 6 = -6 + 6 2x² + 3x + 1 = 0 The equation becomes to 2x² + 3x + 1 = 0, which is in standard form. In the equation 2x² + 3x + 1 = 0, a = 2, b = 3, and c = 1. When b = 0, in the equation ax² + bx + c = 0, it results to a quadratic equation of the form ax² + c = 0. Examples: Equation such as x² + 5 = 0, 2x² + 7 = 0, and 16x² - 9 = 0 are quadratic equations of the form ax² + c = 0. In each equation, the value of b = 0. 4 Activity 1. Quadratic or Not Quadratic? Directions: Identify which of the following equations are quadratic and which are not. 1) x (x2 + 3) = 8 2) x2 – 1 = 0 3) x2 = 16 4) (4 - x)2 = 3x 5) (x2 + 2)2 – 5 = 0 6) (2x - 5) + 2 = 0 7) 3x2 – 10 = 4x 8) (x - 5)2 = 2 9) 2 (x + 3)2 = 0 10) (x - 1)2 + 6 = 0 11) x(x + 3)2 = 7 12) 4 = 2 (x + 1)2 13) 5 (3x + 2) = 4 14) 3x2 + 1 x 15) x2 + 3 5 = x2 =0 Activity 2. Set Me to Your Standard Write each quadratic equation in standard form, ax² + bx + c = 0, then identify the values of a, b, and c. 1. 3x + 2x² = 7 2. 5 + 2x² = 6x 5 3. (x + 3) (x + 4) = 0 4. (2x + 7) (x - 1) =0 5. 2x (x - 3) = 15 A quadratic equation in one variable is a mathematical sentence of degree two that can be written in the following standard form, ax² + bx + c = 0. Where a, b, and c are real numbers and a ≠ 0. In the equation, ax² is the quadratic term, bx is the linear term, and c is the constant term. I. Put a check on the blank if the equation is quadratic or not. 1) 5 (x - 7) = 0 2) 15 = x2 3) (x + 2)2 = x2 4) 3x (x + 2)2 = 5 5) x (2x + 5)2 = 0 6) x [(2 + x)2] = 0 7) 2 [(4x - 1)2] = 8 8) x (x + 3) = 6 9) 5 (x - 5) = 2 10) x (3x2 – 2x + 1) = 0 6 II. Write each equation in the form of ax2 + bx + c = 0 and identify the value of a, b, and c. Equation In the form of ax2 + bx + c =0 Values of a b c 1) 2x2 - 6x = 3x 2) 2 (3x + 1)2 = 0 3) 5 = (4x + 1)2 + 6 4) 7 (2x2 - 1) = 5x 5) 2 (3x + 5) = x2 6) 2x (3x - 3) + 7 = 0 7) 3 (x – 2)2 = 6 8) 2x – 3 = 2(x – 1)2 9) 3x + 5 = 4(x + 1)2 10) 7x + 1 = (2x - 1)2 11) 2x2 = (x + 3)2 12) 8x = 4(x2 + 2x) 13) 6 = (3x + 1)2 14) (3x + 1)2 = 4(x – 1)2 15) 8x2 + 3 = - 4x Directions: Choose the letter of the correct answer. 1. Which of this equation describes a quadratic equation? A. 2(x + 3)2 = 0 B. y + 3x2 = 0 C. 3x – 2 = 0 D. (x + 3) + 8 = 0 2. In the equation 5 - 3x2 = 6x, what is the value of c? A. 5 B. 6 C. -2 7 D. -5 3. Write the equation (2x + 7) (x + 1) = 0 In the standard form. A. 2x2 + 9x – 7 = 0 C. 2x2 + 5x – 7 = 0 B. B. 2x2 + 9x + 7 = 0 D. 2x2 - 5x + 7 = 0 4. In the quadratic equation ax2 + bx + c = 0, a ≠ _. A. 1 B. 0 C. 2 D. -1 5. What is the standard form of 3x + 5x2 = -7 A. 5x2 + 3x = -7 C. 5x2 + 3x +7 = 0 B. 3x + 5x2 + 7 = 0 D. 3x + 5x2 = -7 6. A quadratic equation in the form of ax2 + bx + c = 0 where a, b, and c are all real numbers and a ≠ 0 is said to be in _____________. A. quadratic inequality C. factoring B. standard form D. Quadratic formula 7. In the quadratic equation 9x2 + 8x - 8 = 0, what is the value of a? A. 4 B. 9 C. 0 D. −8 8. Which is the quadratic term in the quadratic equation 5x2 - 3x + 4 = 0? A. x2 B. – 3x C. 5x2 D. 4 9. Which of the following is a quadratic equation? A. 2r2 + 4r – 1 C. x2 + 5x - 4 = 0 B. 3t - 7 = 2 D. 2a2 – 7a ≥ c = 0 10. A mathematical sentence of degree 2 that can be written in the form ax2+bx+c = 0 where a, b, and c are real numbers and a is ≠ 0. A. Quadratic equation C. Linear equation B. Quadratic inequality D. Linear inequality 8 Pretest Activity 1 1. A 1 Not Quadratic 2. C Post Test 2 Quadratic Check Your Understan ding 3. C 3 Quadratic I. 3. B 4. B 4 Quadratic 1. 4. B 5. B 5 Not Quadratic 2. ✓ 5. C 6. C 6 Not Quadratic 3. 6. B 7. B 7 Quadratic 4. 7. B 8. C 8 Quadratic 5. 8. C 9. C 9 Quadratic 6. 9. C 10. A 10 Quadratic 7. ✓ 10. A 11 Not Quadratic 8. ✓ 12 Quadratic 9. ✓ 13 Not Quadratic 10. 14 Not Quadratic 15 Quadratic Activity 2 1) 2x2 + 3x – 7 = 0; a= 2, b = 3, c = -7 2) 2x2 – 6x + 5 = 0; a = 2, b = 6, c = 5 3) x2 + 7x + 12 = 0; a = 1, b = 7, c = 12 4) 2x2 + 5x – 7 = 0; a = 2 5) 2x2 – 6x – 15 = 0; a = 2, b = -6, c = 15 9 1. A 2. A II. Check Your Understanding Equation 1) 2x2 - 6 = 3x 2) 2 (3x + 1)2 = 0 Values of In the form of ax2 + bx + c = 0 a b c 2x2 – 3x – 6 = 0 2 -3 -6 18 12 2 18x2 + 12x + 2= 0 3) 5 = (4x + 1)2 + 6 16x2 + 8x + 2 = 0 16 8 2 4) 7 (2x2 - 1) = 5x 14x2 – 5x – 7 = 0 14 -5 -7 5) 2 (3x + 5) = x2 x2 – 6x – 10 = 0 1 -6 -10 6) 2x (3x - 3) + 7 = 0 6x2 – 6x + 7 = 0 6 -6 7 7) 3 (x – 2)2 = 6 3x2 – 12x + 6 = 0 3 -12 6 8) 2x – 3 = 2(x – 1)2 2x2 – 6x + 5 = 0 2 -6 5 9) 3x + 5 = 4(x + 1)2 4x2 + 5x – 1 = 0 4 5 -1 10) 7x + 1 = (2x - 1)2 4x2 – 11x = 0 4 -11 0 11) 2x2 = (x + 3)2 X2 - 6x – 9 = 0 1 -6 -9 4x2 = 0 4 0 0 13) 6 = (3x + 1)2 9x2 + 6x – 5 = 0 9 6 -5 14) (3x + 1)2 = 4(x – 1)2 5x2 + 14x - 3 = 0 5 14 -3 15) 8x2 + 3 = - 4x 8x2 + 4x + 3 = 0 8 4 3 12) 8x = 4(x2 + 2x) References Learner’s Material for Mathematics -Grade 9, pages 11 - 17 E-MATH (Worktext in Mathematics 9), page 82 10