Year 5 INVESTIGATION - Sum and Product of

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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.
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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.
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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
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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
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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:
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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 
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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.
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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
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