1. Studying Solutions of Quadratic Equations
Section 11.3
Studying Solutions of Quadratic Equations

2. Idea of 11.3 Section 11.2 showed us how to find the answers.
Section 11.3 we show us what type of answers we will have.
The types of answers are
Repeated rational number
Two distinct rational numbers.
Two irrational conjugates
Two complex conjugates
We will determine the type of answer by looking at the determinant. The determinant is b² – 4ac.

3. Repeated Rational Number
The answers will be a repeated rational number if b² - 4ac = 0

4. Example Determine the types of answers given 4x² – 12x + 9 = 0
Therefore a = 4 b = -12 c = 9
b² – 4ac
(-12)² – 4(4)(9) = 144 – 144 = 0
Therefore this equation will have one rational number that will be repeated.

5. Two distinct rational numbers.
The answers will be two distinct rational numbers if b² - 4ac is positive and a perfect square.

6. Example Determine the types of answers given 4m² + 7m = 0
Therefore a = 4 b = 7 c = 0
b² – 4ac
(7)² – 4(4)(0) = 49 – 0 = 49
49 is positive and a perfect square, therefore this equation will have two distinct rational numbers.

7. Two Irrational Conjugates
The answers will be two irrational conjugate numbers if b² - 4ac is positive but not a perfect square

8. Example Determine the types of answers given 10t² - t - 2 = 0
Therefore a = 10 b = -1 c = -2
b² – 4ac
(-1)² – 4(10)(-1) = = 41
41 is positive, but is not a perfect square, therefore this equation will have two distinct irrational conjugates.

9. Two Complex Conjugates
The answers will be two complex conjugate numbers if b² - 4ac is negative

10. Example Determine the types of answers given 3x² + 5 = -7x
Therefore a = 3 b = 7 c = 5
b² – 4ac
(7)² – 4(3)(5) = 49 – 60 = -11
-11 is a negative number, therefore this equation will have two complex conjugates.

11. How to find the equation
How to find the equation given the solutions.
Start with the answers equal to the variable.
Write them as factors.
Combine the factors through multiplication.
FOIL
Simplify
Equation

12. Graphs and Discriminants
The discriminant can help you create the graph.
How?

13. Example Find the equation given the answers -6 and 3 x = -6 or x = 3
(x + 6)(x – 3) = 0*0
X² – 3x + 6x - 18 = 0
X² + 3x - 18 = 0

14. Example
Find the equation given the answers 3, -1, 0

15. Homework
Section 11.3
# 7, 9, 13, 17, 29, 35, 53, 45, 56