October 01, 2014
Lesson 1.13 Sum and Product of the Roots
We will need to write a quadratic equation after we are given the
roots. You have seen this before.
Example 1
Write the quadratic equation that has roots at 1 and 3
Write the roots as x = ...
x2 - 4x + 3 = 0
came from
Work backwards. Consider
(x - 1)(x - 3) = 0
came from
the steps that must have
x - 1 = 0 or x - 3 = 0
come before. Here, start at
came from
the bottom and work up.
x = 1 or x = 3
So, the quadratic equation with roots 1 and 3 is x2 - 4x + 3 = 0
October 01, 2014
Writing a Quadratic Equation from Given Information
We know from factoring a trinomial that the "c" comes from
multiplying (product) and the "b" comes from adding (sum) or
subtracting.
Formulas for the Sum of the Roots and the Product of the Roots
Sum =
Product =
Example 2
Find the sum of the roots AND the product of the roots of the
equation 3 = 2x2 - 6x.
Make sure the equation is equal to zero first!
(Work the problem out on your own using the two formulas above)
This means that when you add the two roots together, whatever
they may be, they will add to 3. When you multiply them the
product will be - 3 .
2
October 01, 2014
Example 3
Write an equation in which the sum of the roots equal ¾ and the
product of the roots equal -2
Our goal is to determine a, b, and c for our quadratic equation.
Sum
sum = -b
a
Product
product = ca
Since the sum of the roots is a fraction, let a = 4 (the denominator)
for both the sum and the product.
3 = -b
4 4
-2 = c
3 = -b
-3 = b
-8 = c
4
So, a = 4 (the denominator), b = -3 (from the sum formula),
and c = -8 (from the product formula)
Then the equation is 4x2 - 3x - 8 = 0
October 01, 2014
Example 4
Write a quadratic equation has the roots (1 + 3i) and (1 - 3i)
First, determine the sum of the roots: (1 + 3i) + (1 - 3i) = 2
Second, determine the product of the roots:
(1 + 3i)(1 - 3i) = 1 - 9i2
= 1 - 9(-1)
=1+9
= 10
So, the sum of the roots is 2, and the product of the roots is 10.
Sum
sum = -b
a
Product
product = ca
Since the sum of the roots is not a fraction, let a = 1 for both the sum
and the product.
2 = -b
10 = c
2 = -b
-2 = b
10 = c
1
1
So, a = 1 (because there was no denominator), b = -2 (from the sum
formula), and c = 10 (from the product formula)
Then the equation is 1x2 - 2x + 10 = 0
October 01, 2014
Classwork for Thursday, 10.2.14
Page 223: Start with the circled questions