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U4.7 Complex Numbers and The Quadratic Formula

U4.7 Complex Numbers and The Quadratic Formula. What can you tell me about this graph????. Warm-up. Diff of Sq: (2x+y)(2x-y). Factor completely 4x 2 -y 2 Factor completely 36x 2 -12x+1 Factor completely c 4 +8c 2 +16 Factor completely 16x 3 +54 Factor completely 8x 3 +8x 2 -48x.

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U4.7 Complex Numbers and The Quadratic Formula

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  1. U4.7 Complex Numbers and The Quadratic Formula What can you tell me about this graph????

  2. Warm-up Diff of Sq: (2x+y)(2x-y) • Factor completely 4x2-y2 • Factor completely 36x2-12x+1 • Factor completely c4+8c2+16 • Factor completely 16x3+54 • Factor completely 8x3+8x2-48x Perfect Sq Trinomial a≠1: (6x-1)2 Perfect Sq Trinomial a=1: (c2+4)(c2 +4) = (c2+4)2 GCF & Sum of Cubes: 2(8x3+27)=2(2x+3)(4x2-6x+9) GCF & Trinomial a = 1: 8x(x2+x-6)=8x(x+3)(x-2)

  3. Imaginary and Complex Numbers • Objective: • To simplify square roots containing negative radicands • To solve quadratic equations that have pure imaginary solutions • To add, subtract, multiply and simplify complex numbers

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

  5. How many Real Solutions? NO Real 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

  6. 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.

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

  8. Examples 1-2 1. Simplify 2. Simplify Four times through the pattern and 1 into the next pattern.

  9. Practice 1-2… 1. Simplify 2. Simplify

  10. Steps for Solving Quadratics without a linear term. • Isolate the Squared Term • Take the square root of both sides of the equation to find x. • Remember to include the ± when you take the square root of both sides of the equation • Simplify to find the values of x that will be the solutions.

  11. Example 3 No Linear Term! Solve: x²+ 16 = 0

  12. Practice 3-4

  13. Examples 4-5 Simplify the expression 4. (-3i)(8i) 5.

  14. Practice what you learned 5. 6.

  15. 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.

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

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

  18. Simplify To add/subtract complex numbers : combine like terms. Example 6: (8 – 5i) + (2 + i) • Practice 7-9 • (4 – i) + (3 + 3i) • (9 – 6i) – (12 + 2i) • 9. (4 + 7i) – (2 + 3i) 7+2i -3-8i 2+4i

  19. Simplify Multiply complex numbers: Distribute or FOIL Ex 7: (8+5i)(2 – 3i) Ex 8: (3-4i)²

  20. Practice 10. (7- 5i)(4 – 4i) 11. 9i(3 - 6i) 12. (8-2i)² 8 - 48i 54+27i 60 - 32i

  21. Complex conjugates • complex conjugates (a + bi) and (a - bi) [Add this information to your notes] • The product of complex conjugates is always a real number. • You can use complex conjugates to remove the imaginary unit i from the denominator of a fraction.

  22. Simplify Step 1. Multiply the numerator and denominator by the conjugate. Step 2. Simplify. • Ex 9

  23. Simplify • Ex 10

  24. Practice 13. 14.

  25. 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.

  26. Discriminant Graphic Organizer

  27. Find the discriminant. Give the number and type of solutions of the equation. Ex 12: Ex 13: Ex 14: 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

  28. 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).

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

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

  31. Example 15 • Solve using the Quadratic formula

  32. Example 16 Simplify the formula • Solve using the Quadratic formula Standard Form Identify the values of a, b and c Plug Values into the Quadratic Formula Write the Solution(s) Simplify under the radical

  33. Example 17 • Solve using the Quadratic Formula imaginary

  34. Practice 15 • Solve using the Quadratic formula imaginary

  35. AbsentStudentsNote Page 3 attachments for today’s lesson. 1. Notes page 2. Complex # System Graphic Organizer 3. Discriminant Graphic organizer

  36. Complex Number Graphic Organizer

  37. Discriminant Graphic Organizer

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