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Algebra 2: Unit 5 Continued

Algebra 2: Unit 5 Continued. Factoring Quadratic expression. Factors. Factors are numbers or expressions that you multiply to get another number or expression. Ex. 3 and 4 are factors of 12 because 3x4 = 12. Factors. What are the following expressions factors of?

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Algebra 2: Unit 5 Continued

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  1. Algebra 2: Unit 5 Continued Factoring Quadratic expression

  2. Factors Factors are numbers or expressions that you multiply to get another number or expression. Ex. 3 and 4 are factors of 12 because 3x4 = 12

  3. Factors What are the following expressions factors of? 1. 4 and 5? 2. 5 and (x + 10) 3. 4 and (2x + 3) 4. (x + 3) and (x - 4) 5. (x + 2) and (x + 4) 6. (x – 4) and (x – 5)

  4. GCF One way to factor an expression is to factor out a GCF or a GREATEST COMMON FACTOR. EX: 4x2+ 20x – 12 EX: 9n2 – 24n

  5. Try Some! Factor: 9x2 +3x – 18 7p2 + 21 4w2 + 2w

  6. Factors of Quadratic Expressions When you multiply 2 binomials: (x + a)(x + b) = x2 + (a +b)x + (ab) This only works when the coefficient for x2 is 1.

  7. Finding Factors of Quadratic Expressions When a = 1: x2+ bx + c Step 1. Determine the signs of the factors Step 2. Find 2 numbers that’s product is c, and who’s sum is b.

  8. Sign table!

  9. Examples Factor: 1. X2 + 5x + 6 2. x2 – 10x + 25 3. x2 – 6x – 16 4. x2 + 4x – 45

  10. Examples Factor: 1. X2 + 6x + 9 2. x2 – 13x + 42 3. x2 – 5x – 66 4. x2 – 16

  11. More Factoring! When a does NOT equal 1. Steps Slide Factor Divide Reduce Slide

  12. Example! Factor: 1. 3x2 – 16x + 5

  13. Example! Factor: 2. 2x2+ 11x + 12

  14. Example! Factor: 3. 2x2+ 7x – 9

  15. Try Some! Factor 1. 5t2 + 28t + 32 2. 2m2 – 11m + 15

  16. March 20th Warm Up Find the Vertex, Axis of Symmetry, X-intercept, and Y-intercept for each: y = x2 + 8x + 9 y= 2(x – 3)2 + 5

  17. Quadratic Equations

  18. Quadratic Equation Standard Form of Quadratic Function: y = ax2 + bx + c Standard Form of Quadratic Equation: 0 = ax2 + bx + c

  19. Solutions A SOLUTION to a quadratic equation is a value for x, that will make 0 = ax2 + bx + c true. A quadratic equation always have 2 solutions.

  20. 5 ways to solve There are 5 ways to solve quadratic equations: Factoring Finding the Square Root Graphing Completing the Square Quadratic Formula

  21. SOLVING BY FACTORING

  22. Factoring Solve by factoring; 2x2 – 11x = -15

  23. Factoring Solve by factoring; x2 + 7x = 18

  24. Factoring Solve by factoring; 1. 2x2 + 4x = 6 2. 16x2 – 8x = 0 3. x2 – 9x + 18 = 0

  25. Solving by Finding Square Roots For any real number x; X2 = n x = Example: x2 = 25

  26. Solve Solve by finding the square root; 5x2 – 180 = 0

  27. Solve Solve by finding the square root; 4x2 – 25 = 0

  28. Try Some! Solve by finding the Square Root: 1. x2 – 25 = 0 2. x2 – 15 = 34 3. x2 – 14 = -10 4. (x – 4)2 = 25

  29. Quadratic Equations Solving by Graphing

  30. Warm Up March 21st A model for a company’s revenue is R = -15p2 + 300p + 12,000 where p is the price in dollars of the company’s product. What PRICE will maximize the Revenue? What is the maximum revenue? Convert to vertex form: y = 2x2 + 6x - 8

  31. 5 ways to solve There are 5 ways to solve quadratic equations: Factoring Finding the Square Root Graphing Completing the Square Quadratic Formula

  32. Solving by Graphing For a quadratic function, y = ax2 +bx + c, a zero of the function, or where a function crosses the x-axis, is a solution of the equations ax2 + bx + c = 0

  33. Examples Solve x2 – 5x + 2 = 0

  34. Examples Solve x2 + 6x + 4 = 0

  35. Examples Solve 3x2 + 5x – 12 = 8

  36. Examples Solve x2 = -2x + 7

  37. Complex Numbers

  38. Quick Review Simplifying Radicals If the number has a perfect square factor, you can bring out the perfect square. EX:

  39. Try Some

  40. Try this: Solve the following quadratic equations by finding the square root: 4x2 + 100 = 0 What happens?

  41. Complex Numbers

  42. Imaginary Number:i The Imaginary number This can be used to find the root of any negative number. EX

  43. Properties of i This pattern repeats!!

  44. Graphing Complex Number

  45. Absolute Values

  46. Operations with Complex Numbers The Imaginary unit, i, can be treated as a variable Adding Complex Number EX: (8 + 3i) + ( -6 + 2i)

  47. Try Some! 7 – (3 + 2i) (4 – 6i) + 3i

  48. Operations with Complex Numbers Multiplying Complex Numbers:Example: (5i)(-4i) Example: (2 + 3i)(-3 + 5i)

  49. Try Some! (6 – 5i)(4 – 3i) (4 – 9i)(4 + 3i)

  50. Now we can SOLVE THIS! Solve 4x2 + 100 = 0

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