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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)

  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. Box and Slide Method 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. You Try! Factor 1. 5t2 + 28t + 32 2. 2m2 – 11m + 15

  16. Quadratic Equations

  17. 5 ways to solve There are 5 ways to solve quadratic equations: Factoring Finding the Square Root Graphing Completing the Square Quadratic Formula

  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. Note: A quadratic equation always have 2 solutions.

  20. SOLVING BY FACTORING

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

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

  23. Solve by Factoring: 1. 2x2 + 4x = 6 2. 16x2 – 8x = 0

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

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

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

  27. 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

  28. Quadratic Equations Solving by Graphing

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

  30. Examples Solve x2 – 5x + 2 = 0

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

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

  33. Examples Solve x2 = -2x + 7

  34. Complex Numbers

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

  36. You Try!

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

  38. Complex Numbers

  39. Imaginary Number:i The Imaginary number This can be used to find the root of any negative number. For example:

  40. Properties of i

  41. Graphing Complex Number

  42. Absolute Values

  43. Absolute Values

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

  45. You Try! 7 – (3 + 2i) (4 – 6i) + 3i

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

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

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

  49. Completing the Square

  50. 5 ways to solve There are 5 ways to solve quadratic equations: Factoring Finding the Square Root Graphing Completing the Square Quadratic Formula

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