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SUM and PRODUCT of the ROOTS of a QUADRATIC EQUATION MYP 5 – Math Investigation Assessment: Criterion B Rubric for Criterion B – Investigating Patterns Level of Descriptor Achievement 0 The student does not reach a standard described by any of the descriptors given below. 1–2 The student applies, with some guidance, mathematical problem solving techniques to recognize simple patterns. 3–4 The student selects and applies mathematical problem solving techniques to recognize patterns, and suggests relationships or general rules. 5–6 The student selects and applies mathematical problem solving techniques to recognize patterns, describes them as relationships or general rules, and draws conclusions consistent with findings. 7–8 The student selects and applies mathematical problem solving techniques to recognize patterns, describes them as relationships or general rules, draws conclusions consistent with findings, and provides justifications or proofs. Page 1 of 8 Background Information: In this investigation you are going to be solving quadratic equations and finding a pattern in your solutions. For the purpose of this investigation, your solutions are called roots i.e. Find the roots of the following quadratic equation, then find the sum of the roots and the product of the roots: x 2 3x 2 0 Working: ( x 2)( x 1) 0 x 2, x 1 -2 and -1 are called the roots of the quadratic equation Sum of roots: -2 + -1 = -3 Product of roots: (-2)(-1) = 2 Sum of roots just means to add your roots (solutions), and product of roots means to multiply your roots (solutions). You’ll do this for each quadratic equation. Page 2 of 8 INVESTIGATING THE ROOTS OF A QUADRATIC EQUATION OBJECTIVES OF THE INVESTIGATION - To recognize the patterns existing between the coefficients of a quadratic equation and the roots of the equation. - To apply the patterns you discover. PART I: INVESTIGATING THE PATTERN 1. Solve each of the following quadratic equations using the method of your choice then find the sum (add) and the product (multiply) of the roots (solutions) for each equation. 1) x2 5x 6 0 2) x 2 3x 18 0 3) x2 11x 30 0 4) x 2 3x 2 0 5) x2 5x 4 0 6) 6 x2 5x 1 0 Page 3 of 8 7) 3x 2 5 x 2 0 8) 4 x 2 3x 10 0 9) 3 y 2 10 y 8 0 10) 6 x2 x 2 0 Page 4 of 8 2. Organize your results and describe mathematically or in words what you think the relationship is between the numerical coefficients and the roots of a quadratic equation. 3. Use the example x 2 7 x 18 0 and test your theory. If you are wrong, re-examine your theory. PREDICTION : Sum of roots : Product of roots: PROOF: 4. Use the example 2 x 2 5 x 2 0 and test your theory. If you are wrong, re-examine your theory. PREDICTION : Sum of roots : Product of roots: PROOF: Page 5 of 8 PART II: FINDING THE GENERAL RULE 1. Given the quadratic equation ax a) the sum of the roots 2 bx c 0 , state a general expression (formula) for: b) the product of the roots PART III: APPLICATIONS OF THE RULE 1. Using the rule that you developed above, find the sum and product of the roots of each of the following equations. DO NOT SOLVE THE EQUATIONS. a) x 2 5 x 12 0 b) 3 x 2 5 x 12 c) 6 x 2 18 0 d) 4 x 2 x 40 0 2. Determine if each of the following quadratic equations have the given solutions without actually solving the equation. Explain your answers and/or demonstrate why or why not. a) x2 6 x 8 0; 2, 4 b) 2 5 30 x 2 13 x 10 0; , 5 6 Page 6 of 8 3. Use what you have learned in this investigation to find the quadratic equation with the given roots. Explain your reasoning and/or clearly demonstrate what you are doing. a) 4. 5, 7 If –6 is one root of the equation b) 4 x 2 3 x c 0 , find c 1 3,1 3 and the other root. Show your work/reasoning clearly. Page 7 of 8 PART IV: PROVING YOUR RESULTS 1. A mathematical proof is a demonstration that a rule applies in the general case. Given any quadratic ax that the relationship(s) you have discovered between the numerical coefficients and the roots is/are true. 2 bx c 0 , prove Page 8 of 8