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I. Model Problems.
II. Practice
III. Challenge Problems
IV. Answer Key
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Sum and Product of Roots:
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Sum and Product Rule of Quadratics
For any quadratic equation
The sum of the roots of the equation is
:
. The product of the roots of the equation is .
I. Model Problems
In this example you will find the sum and product of the roots of a quadratic equation.
Example 1: Find the sum and product of the roots of
Identify a, b, and c.
Sum of the roots.
Substitute and simplify.
Product of the roots.
Substitute and simplify.
Answer:
In this example you will use the sum and product rule to determine if two values are the
roots of a quadratic equation.
Example 2: Are 7 and 2 the roots of
?
Identify a, b, and c.
Find the sum of the potential roots.
Find the actual sum using the sum and
product rule.
Compare.
Answer: 7 and 2 are not the roots.
You could have checked the product:
and arrived at the same answer. Both
the sum and product must check for the potential roots to be the roots.
In this example you will find a quadratic equation with the given roots.
Find a quadratic equation with the roots
.
Find the sum of the roots.
Find the product of the roots.
We know
Rewrite as fractions.
We want the denominators of the fractions to be the same so that a is the same for both
equations. In this case a is one for both. Then we can identify a, b, and c.
Identify a, b, and c.
Answer:
(there are many quadratics that have the given roots, but once
the GCF is factored this will be the quadratic e.g.
).
In this example you will find the missing root of an equation.
Find the missing root of
, if one root is
. Let
the roots of the equation.
First identify what you know.
We know a and b so use the sum rule.
Sum rule.
Substitute.
Solve.
Answer: the missing root is
.
represent
II. Practice
Find the sum and product of the roots of the given quadratic equation.
1.
2.
3.
4.
Use the sum and product rule to determine if the two given values are the roots of
the quadratic equation.
5. Are
the roots of
?
6. Are
the roots of
?
7. Are
8. Are
the roots of
?
the roots of
?
Find a quadratic equation for the given roots.
9.
10.
11.
12.
13.
14.
15.
16.
Find the missing root.
17. Given
is a root of
18. Given
is a root of
19. Given
is a root of
20. Given
21. Given
.
.
.
is a root of
.
is a root of
.
III. Challenge Problems
22. Find the polynomial with the roots
23. Find the polynomial with the roots
24. Find the missing root given
.
.
is a root of
25. Find the missing root given is a root of
.
.
IV. Answer Key
1. sum:
2. sum:
product:
product:
3. sum:
product:
4. sum:
product:
5. no
6. no
7. yes
8. no
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.