1. 5.8 Quadratic Formula

2. For quadratic equations written in standard form, the roots can be found using the following formula: This is called the Quadratic Formula. It is one of the more important formulas that you will learn this year. Memorize it!!!!

3. Find the roots: a = 1 b = 9 c = 14

4. Find the roots: a = 5 b = -8 c = 1

5. Find the roots: a = -3 b = -5 c = 5

6. Find the roots: a = 3 b = -6 c = 3

7. Find the roots: a = 4 b = 2 c = 1

8. Find the roots: a = 7 b = -5 c = 10

9. Discriminants b 2 – 4ac is called the discriminant. It will allow you to quickly determine how many roots a particular quadratic equation has.

10. Discriminants In the previous examples, there were three cases. Case #1: 2 Real solutions: The normal situation for the quadratic formula is that it gives two real numbers as the solutions. When the discriminant is positive, there will be 2 real solutions. There are 2 real solutions because of what is inside the square root is positive. When you add and subtract a positive real number, you get 2 different real answers.

11. Discriminants Case #2: 1 Real solution: Sometimes both the solutions are the same. It is then said that there is only one distinct solution. When the discriminant is zero, there will be 1 real solution. There is 1 real solution because of what is inside the square root is zero. When you add and subtract a zero, you get the same answer both times.

12. Discriminants Case #3: 2 Complex solutions: Sometimes both the solutions are complex. It is then said that there are no real solutions. When the discriminant is negative, there will be 2 complex solutions. There are 2 complex solutions because of what is inside the square root is negative. (This will become an i.) When you add and subtract an imaginary number, you get 2 different complex numbers.

13. Use the discriminant to determine how many solutions there will be for each equation. x 2 + 12x + 3 = 0 4x 2 - 12x + 9 = 0 = 132 Positive Number 2 Real Solutions = 0 Zero 1 Real Solution

14. Use the discriminant to determine how many solutions there will be for each equation. 7x 2 - x + 2 = 0 24x 2 - 14x - 5 = 0 = -55 Negative Number 0 Real Solutions (2 Complex Solutions) = 676 Positive Number 2 Real Solutions

15. Use the discriminant to determine how many solutions there will be for each equation. 16x 2 + 40x + 25 = 0 3x 2 - 21 = 0 = 0 Zero 1 Real Solution = 252 Positive Number 2 Real Solutions

16. Use the discriminant to determine how many solutions there will be for each equation. Negative 0 Real Solutions (2 Complex Solutions) Not a quadratic!!! Don’t use discriminant!!!