1. Simplifying a Radical Review Simplify each radical and leave the answer in exact form. 1. 2. 3.

2. How many Real Solutions? 2 Complex Solutions Two Real Solutions One Real Solutions How did you determine your answer? Looking at the number of times it touches or crosses the x axis

3. Imaginary unit Not all Quadratic Equations have real-number solutions. To overcome this problem, mathematicians created an expanded system of numbers using the imaginary unit The imaginary number is use to write the square root of any negative number.

5. Definition For any positive real number b, Modular 4 Pattern

6. Example 1 Solve: x²+ 16 = 0

7. Complex numbers Expression that contains a real number and a pure imaginary number in the form (a + bi) 5 + 2i 5 is the real number 2i is the imaginary part.

8. Complex Number System Graphic Organizer Is every real number a complex number? Yes Is every imaginary number a complex number? Yes Is every Complex number a real number? NO

9. Life is complex. It has real and imaginary components.

10. Discriminant The expression b²- 4ac is called the Discriminant of the equation ax² + bx + c = 0 From the discriminant we can tell the nature and number of solutions for any given quadratic function.

11. Discriminant Graphic Organizer Type OneType TwoType Three Value of the Discriminant: b 2 - 4ac >0b 2 - 4ac = 0b 2 - 4ac < 0 Number and Type of Solutions: Two Real Solutions One Real Solution Two Imaginary Solutions Number of Intercepts: Two x-intercept One x-intercept No x-intercept Graph of Example:

12. Find the discriminant. Give the number and type of solutions of the equation. Ex 2: Ex 3: Ex 4: Disc b² - 4ac= (-8)²- 4(1)(17)= -4 -4<0 so Two imaginary solutions Disc (-8)²- 4(1)(16)= 0 0=0 so One real solutions Disc (-8)²-4(1)(15) = 4 4>0 so Two real solutions

13. Quadratic Formula Objective: –To use the quadratic formula to find the solutions. Let a, b, and c be real numbers such that a ≠ 0. Use the following formula to find the solutions of the equation ax² + bx+ c = 0 (Standard Form).

14. Can you Sing it? X equals the opposite of b plus or minus the square root of b squared minus four AC all over 2 A. Yes you Can! Pop Goes The Weasel!

15. Parts of the Quadratic Formula ax² + bx + c = 0 Quadratic Formula   Discriminant x-value of the Vertex What kind of solutions and how many? Method to find solutions of a quadratic equation.

16. Example 5 Solve using the Quadratic formula

17. Example 6 Solve using the Quadratic formula Standard Form Identify the values of a, b and c Plug Values into the Quadratic Formula Simplify under the radical Simplify the formula Write the Solution(s)

18. Example 7 Solve using the Quadratic Formula imaginary

19. Practice 8 Solve using the Quadratic formula imaginary