1. Notes Packet 10: Solving Quadratic Equations by the Quadratic Formula

2. THE QUADRATIC FORMULA 1.The general quadratic equation 2.This is the quadratic formula! 3.Just identify a, b, and c then substitute into the formula.

3. Identifying a, b, and c From the quadratic equation, ax 2 +bx+c=0 The quadratic term is a The linear term is b The constant is c Example: x 2 + 2x – 5 = 0 a = 1 b = 2 c = -5

4. WHY USE THE QUADRATIC FORMULA? The quadratic formula allows you to solve ANY quadratic equation, even if you cannot graph it. An important piece of the quadratic formula is what ’ s under the radical: b 2 – 4ac This piece is called the discriminant.

5. WHY IS THE DISCRIMINANT IMPORTANT? The discriminant tells you the number and types of answers (roots) you will get. The discriminant can be +, –, or 0 which actually tells you a lot! Since the discriminant is under a radical, think about what it means if you have a positive or negative number or 0 under the radical.

6. WHAT THE DISCRIMINANT TELLS YOU! Value of the DiscriminantNature of the Solutions Negative2 imaginary solutions Zero1 Real Solution Positive – perfect square2 Reals- Rational Positive – non-perfect square 2 Reals- Irrational

7. Example #1 Find the value of the discriminant and describe the nature of the roots (real,imaginary, rational, irrational) of each quadratic equation. Then solve the equation using the quadratic formula) 1. a=2, b=7, c=-11 Discriminant = Value of discriminant=137 Positive-NON perfect square Nature of the Roots – 2 Reals - Irrational

8. Example #1- continued Solve using the Quadratic Formula

9. Solving Quadratic Equations by the Quadratic Formula Try the following examples. Do your work on your paper and then check your answers.

10. Complex Numbers

11. Definition of pure imaginary numbers: Any positive real number b, where i is the imaginary unit and bi is called the pure imaginary number.

12. Definition of pure imaginary numbers: i is not a variable it is a symbol for a specific number

13. Simplify each expression.

14. Remember Simplify each expression. Remember

16. Simplify. To figure out where we are in the cycle divide the exponent by 4 and look at the remainder.

17. Simplify. Divide the exponent by 4 and look at the remainder.

18. Simplify. Divide the exponent by 4 and look at the remainder.

19. Simplify. Divide the exponent by 4 and look at the remainder.