1. Quadratic Equations
Irrational Roots and Sum & Product of the Roots

2. Solve for x by using the quadratic formula:
Aim: Use the sum and product of the roots in order to write a quadratic equation.
Solve for x by using the quadratic formula:
a. What is the sum of its roots? = -7
b. What is the product of its roots?
-3  -4 = 12

3. Sometimes you are given information backwards!
Aim: Use the sum and product of the roots in order to write a quadratic equation.
Sometimes you are given information backwards!
You may be given information and asked to write a quadratic equation.
Remember – it’s all about finding which factors of ac add to b.

4. Sum and Product of Roots for a Quadratic Equation
Aim: Use the sum and product of the roots in order to write a quadratic equation.
Sum and Product of Roots for a Quadratic Equation
The sum of the roots is:

5. The product of the roots is:
Aim: Use the sum and product of the roots in order to write a quadratic equation.
The product of the roots is:

6. Sum of roots = Product of roots =
Aim: Use the sum and product of the roots in order to write a quadratic equation.
Sum of roots = Product of roots =
Given the equation x2 + x – 20 = 0,
a = 1
b = 1
c =
-20
a. What is the sum of the roots? = = -1
b. What is the product of the roots? = =
-20

7. Sum of roots = Product of roots =
Aim: Use the sum and product of the roots in order to write a quadratic equation.
Sum of roots = Product of roots =
The roots of a quadratic are and
-b = 39
a. What is the sum of the roots?
b = -39
a = 20
b. What is the product of the roots?
c = - 80
c. What is the quadratic equation? 20 x2
-39 x
- 80
= y

8. Sum of roots = Product of roots =
Aim: Use the sum and product of the roots in order to write a quadratic equation.
Sum of roots = Product of roots =
The roots of a quadratic are 5 + 2i and 5 – 2i.
a. What is the sum of the roots?
-b = 10
b = - 10
(5 + 2i) + (5 – 2i) = 10
a = 1
b. What is the product of the roots?
(5 + 2i)  (5 – 2i) =
i – 10i – 4i2
c = + 29
25 – 4(-1) = 29
c. What is the quadratic equation? 1 x2
- 10 x
+ 29
= y

9. Irrational Root Theorem
Aim: Use the sum and product of the roots in order to write a quadratic equation.
Irrational Root Theorem
For a polynomial 𝑦= 𝑎 0 𝑥 𝑛 + 𝑎 1 𝑥 𝑛−1 +…+ 𝑎 𝑛−1 𝑥+ 𝑎 𝑛
If 𝑎 + 𝑏 is a root,
Then 𝑎 − 𝑏 is also a root
Irrationals always come in pairs. Real values do not.
Conjugate: Complex pairs in form 𝑎+ 𝑏 and 𝑎− 𝑏

10. Example of Irrational Root Theorem
Aim: Use the sum and product of the roots in order to write a quadratic equation.
Example of Irrational Root Theorem
1. A polynomial function with roots 𝟐+ 𝟑 . Find all additional roots and degree of polynomial.
𝟐− 𝟑 2
Other root:________
Degree of Polynomial:________
2. A polynomial function with roots −𝟏, 𝟎, − 𝟑 , 𝟏− 𝟓 . Find all additional roots and degree of polynomial.
𝟑 , 𝟏+ 𝟓 6
Other root:________
Degree of Polynomial:________
3. A polynomial function with roots 𝟏− 𝟑 , 𝟐+𝒊. Find all additional roots and degree of polynomials.
1+ 𝟑 , 𝟐−𝒊 4
Other root:________
Degree of Polynomial:________

11. Sum of roots = Product of roots =
Aim: Use the sum and product of the roots in order to write a quadratic equation.
Sum of roots = Product of roots =
One root of a quadratic is
a. What is the sum of the roots?
-b = 6
b = - 6
a = 1
b. What is the product of the roots?
c = + 7
c. What is the quadratic equation? 1 x2
- 6 x
+ 7
= y

12. Sum of roots = Product of roots =
Aim: Use the sum and product of the roots in order to write a quadratic equation.
Sum of roots = Product of roots =
The sum of the roots of a quadratic is , the product of the roots is What is the equation of the quadratic?
-b = 3
c = -2
b = - 3
a = 4
a = 2
( ) 2
a = 4 4 x2
- 3 x
- 2
= y

13. Sum of roots = Product of roots =
Aim: Use the sum and product of the roots in order to write a quadratic equation.
Sum of roots = Product of roots =
Find k such that -3 is a root of x2 + kx – 24 = 0.
Product =
Sum =

14. Polynomials equations greater than quadratics.
Aim: Use the sum and product of the roots in order to write a quadratic equation.
Polynomials equations greater than quadratics.
Write a polynomial function of least degree that has the given zeros.
Roots:______
5, − 3
Other possible roots:______ 3
Degree of Polynomial:________ 3
Then 𝑥=5, 𝑥=− 3 𝑎𝑛𝑑 𝑥= 3
So, 𝑥−5=0, 𝑥+ 3 =0 𝑎𝑛𝑑 𝑥− 3 =0

15. If 𝑥−5=0, 𝑥+ 3 =0 𝑎𝑛𝑑 𝑥− 3 =0 then (𝑥−5)(𝑥+ 3 )(𝑥− 3 )=0 Foil
Aim: Use the sum and product of the roots in order to write a quadratic equation.
If 𝑥−5=0, 𝑥+ 3 =0 𝑎𝑛𝑑 𝑥− 3 =0
then (𝑥−5)(𝑥+ 3 )(𝑥− 3 )=0
Foil
(𝑥−5)( 𝑥 2 −3)=0
Foil again
or distrubute
𝑥 3 −5 𝑥 2 −3𝑥+15=0
If needed
distribute again
or set equal to 𝑦
𝑥 3 −5 𝑥 2 −3𝑥+15=𝑦

16. Other possible roots:________ − 5 , 2
Aim: Use the sum and product of the roots in order to write a quadratic equation.
Write a polynomial function of least degree that has the given zeros: 5 , − 2
Other possible roots:________
− 5 , 2
Degree of Polynomial:________ 4
𝑥= 5 , 𝑥=− 5 , 𝑥=− 2 𝑎𝑛𝑑 𝑥= 2
𝑥− 5 =0, 𝑥+ 5 =0, 𝑥+ 2 =0 𝑎𝑛𝑑 (𝑥− 2 )=0
𝑥− 5 𝑥 𝑥+ 2 (𝑥− 2 )=0
𝑥 2 −5 ( 𝑥 2 −2)=0
( 𝑥 4 −7 𝑥 2 +10)=0
𝑥 4 −7 𝑥 2 +10=𝑦