PREVIEW/SUMMARY OF QUADRATIC EQUATIONS & FUNCTIONS

1. PREVIEW/SUMMARY OF QUADRATIC EQUATIONS & FUNCTIONS QUADRATIC – means second power Recall LINEAR – means first power

2. Click on the Quadratic Method you wish to review to go to those slides Factoring Method – Quick, but only works for some quadratic problems Square Roots of Both Sides – Easy, but only works when you can get in the form of (glob)2 = constant Complete the Square– Always works, but is recommended only when a = 1 or all terms are evenly divisible to set a to 1. Quadratic Formula – Always works, but be sure to rewrite in standard form first of ax2 + bx + c = 0 How to Choose Among Methods 1 – 4? Summary & Hints Quadratic Functions – Summary of graphing parabolas from f(x) = ax2 + bx + c form

3. METHOD 1 - FACTORING • Set equal to zero • Factor • Use the Zero Product Property to solve (Each factor with a variable in it could be equal to zero.)

4. METHOD 1 - FACTORING Any # of terms – Look for GCF factoring first! • 5x2 = 15x 5x2 – 15x = 0 {0, 3} 5x (x – 3) = 0 5x = 0 OR x – 3 = 0 x = 0 OR x = 3

5. Conjugates METHOD 1 - FACTORING Binomials – Look for Difference of Squares 2. x2 = 9 x2 – 9 = 0 {– 3, 3} (x + 3) (x – 3) = 0 x + 3 = 0 OR x – 3 = 0 x = – 3 OR x = 3

6. METHOD 1 - FACTORING Trinomials – Look for PST (Perfect Square Trinomial) 3. x2 – 8x = – 16 x2 – 8x + 16 = 0 {4 d.r.} (x – 4) (x – 4) = 0 x – 4 = 0 OR x – 4 = 0 Double Root x = 4 OR x = 4

7. METHOD 1 - FACTORING Trinomials – Look for Reverse of Foil 4. 2x3 – 15x = 7x2 2x3 – 7x2 –15x =0 {-3/2, 0, 5} (x) (2x2 – 7x – 15) = 0 (x) (2x + 3)(x – 5) = 0 x = 0 OR 2x + 3 = 0 OR x – 5 = 0 x = 0 OR x = – 3/2 OR x = 5

8. PREVIEW/SUMMARY OF QUADRATIC EQUATIONS & FUNCTIONS END OF METHOD 1: FACTORING Click here to return to menu slide

9. METHOD 2 – SQUARE ROOTS OF BOTH SIDES • Reorder terms IF needed • Works whenever form is (glob)2 = c • Take square roots of both sides (Remember you will need a sign!) • Simplify the square root if needed • Solve for x. (Isolate it.)

10. METHOD 2 – SQUARE ROOTS OF BOTH SIDES • x2 = 9 {-3, 3} x =  3 Note  means both +3 and -3! x = -3 OR x = 3

11. METHOD 2 – SQUARE ROOTS OF BOTH SIDES 2. x2 = 18

12. METHOD 2 – SQUARE ROOTS OF BOTH SIDES 3. x2 = – 9 Cannot take a square root of a negative. There are NO real number solutions!

13. METHOD 2 – SQUARE ROOTS OF BOTH SIDES 4. (x-2)2 = 9 {-1, 5} This means: x = 2 + 3 and x = 2 – 3 x = 5 and x = – 1

14. METHOD 2 – SQUARE ROOTS OF BOTH SIDES Rewrite as (glob)2 = c first if necessary. 5. x2 – 10x + 25 = 9 (x – 5)2 = 9 {2, 8} x = 8 and x = 2

15. METHOD 2 – SQUARE ROOTS OF BOTH SIDES Rewrite as (glob)2 = c first if necessary. 6. x2 – 10x + 25 = 48 (x – 5)2 = 48

16. PRACTICE METHOD 2 – SQUARE ROOTS OF BOTH SIDES • x2 = 121 {-11, 11} x =  11 Note  means both +11 and -11! x = -11 OR x = 11

17. PRACTICE METHOD 2 – SQUARE ROOTS OF BOTH SIDES 2. x2 = – 81 Square root of a negative, so there are NO real number solutions!

18. PRACTICE METHOD 2 – SQUARE ROOTS OF BOTH SIDES Rewrite as (glob)2 = c first if necessary. 3. 6x2 = 156 x2 = 26

19. PRACTICE METHOD 2 – SQUARE ROOTS OF BOTH SIDES 4. (a – 7)2 = 3

20. PRACTICE METHOD 2 – SQUARE ROOTS OF BOTH SIDES Rewrite as (glob)2 = c first if necessary. 5. 9(x2 – 14x + 49) = 4 (x – 7)2 = 4/9 {6⅓, 7⅔}

21. PRACTICE METHOD 2 – SQUARE ROOTS OF BOTH SIDES 6.

22. PREVIEW/SUMMARY OF QUADRATIC EQUATIONS & FUNCTIONS END OF METHOD 2: SQUARE ROOTS OF BOTH SIDES Click here to return to menu slide

23. METHOD 3 – COMPLETE THE SQUARE • Goal is to get into the format: (glob)2 = c • Method always works, but is only recommended when a = 1 or all the coefficients are divisible by a • We will practice this method repeatedly and then it will keep getting easier!

24. METHOD 3 – COMPLETE THE SQUARE Example: 3x2 – 6 = x2 + 12x Simplify and write in standard form: ax2 + bx + c = 0 2x2 – 12x – 6 = 0 x2 – 6x – 3 = 0 Set a = 1 by division Note: in some problems a will already be equal to 1.

25. METHOD 3 – COMPLETE THE SQUARE x2 – 6x – 3 = 0 Move constant to other side Leave space to replace it! x2 – 6x = 3 x2 – 6x + 9 = 3 + 9 Add (b/2)2 to both sides This completes a PST! (x – 3)2 = 12 Rewrite as (glob)2 = c

26. METHOD 3 – COMPLETE THESQUARE (x – 3)2 = 12 Take square roots of both sides – don’t forget  Simplify Solve for x

27. PRACTICE METHOD 3 – COMPLETE THE SQUARE Example: 2b2 = 16b + 6 Simplify and write in standard form: ax2 + bx + c = 0 2b2 – 16b – 6 = 0 b2 – 8b – 3 = 0 Set a = 1 by division Note: in some problems a will already be equal to 1.

28. PRACTICE METHOD 3 – COMPLETE THE SQUARE b2 – 8b – 3 = 0 Move constant to other side Leave space to replace it! b2 – 8b = 3 b2 – 8b + 16 = 3 +16 Add (b/2)2 to both sides This completes a PST! (b – 4)2 = 19 Rewrite as (glob)2 = c

29. PRACTICE METHOD 3 – COMPLETE THE SQUARE (b – 4)2 = 19 Take square roots of both sides – don’t forget  Simplify Solve for the variable

30. PRACTICE METHOD 3 – COMPLETE THE SQUARE Example: 3n2 + 19n + 1 = n - 2 Simplify and write in standard form: ax2 + bx + c = 0 3n2 + 18n + 3 = 0 n2 + 6n + 1 = 0 Set a = 1 by division Note: in some problems a will already be equal to 1.

31. PRACTICE METHOD 3 – COMPLETE THE SQUARE n2 + 6n + 1 = 0 Move constant to other side Leave space to replace it! n2 + 6n = -1 n2 + 6n + 9 = -1 + 9 Add (b/2)2 to both sides This completes a PST! (n + 3)2 = 8 Rewrite as (glob)2 = c

32. PRACTICE METHOD 3 – COMPLETE THE SQUARE (n + 3)2 = 8 Take square roots of both sides – don’t forget  Simplify Solve for the variable

33. PRACTICE METHOD 3 – COMPLETE THE SQUARE What number “completes each square”? 1. x2 – 10x = -3 1. x2 – 10x + 25 = -3 + 25 2. x2 + 14 x = 1 2. x2 + 14 x + 49 = 1 + 49 3. x2 – 1x = 5 3. x2 – 1x + ¼ = 5 + ¼ 4. 2x2 – 40x = 4 4. x2 – 20x + 100 = 2 + 100

34. PRACTICE METHOD 3 – COMPLETE THE SQUARE Now rewrite as (glob)2 = c 1. (x – 5)2 = 22 1. x2 – 10x + 25 = -3 + 25 2. x2 + 14 x + 49 = 1 + 49 2. (x + 7)2 = 50 3. x2 – 1x + ¼ = 5 + ¼ 3. (x – ½ )2 = 5 ¼ 4. x2 – 20x + 100 = 2 + 100 4. (x – 10)2 = 102

35. PRACTICE METHOD 3 – COMPLETE THE SQUARE Show all steps to solve. ⅓k2 = 4k - ⅔ k2 = 12k - 2 k2 - 12k = - 2 k2 - 12k + 36 = - 2 + 36 (k - 6)2 = 34

36. PREVIEW/SUMMARY OF QUADRATIC EQUATIONS & FUNCTIONS END OF METHOD 3: COMPLETE THE SQUARE Click here to return to menu slide

37. METHOD 4 – QUADRATICFORMULA • This is a formula you will need to memorize! • Works to solve all quadratic equations • Rewrite in standard form in order to identify the values of a, b and c. • Plug a, b & c into the formula and simplify! • QUADRATIC FORMULA:

38. METHOD 4 – QUADRATICFORMULA Use to solve: 3x2 – 6 = x2 + 12x Standard Form: 2x2 – 12x – 6 = 0

39. METHOD 4 – QUADRATICFORMULA

40. PRACTICE METHOD 4 – QUADRATIC FORMULA Show all steps to solve & simplify. 2x2 = x + 6 2x2 – x – 6 =0

41. PRACTICE METHOD 4 – QUADRATIC FORMULA Show all steps to solve & simplify. x2 + x + 5 = 0

42. PRACTICE METHOD 4 – QUADRATIC FORMULA Show all steps to solve & simplify. x2 +2x - 4 = 0

43. THE DISCRIMINANT – MAKING PREDICTIONS b2 – 4ac is called the discriminant Four cases: 1. b2 – 4acpositive non-square two irrational roots 2. b2 – 4acpositive square two rational roots 3. b2 – 4aczero one rational double root 4. b2 – 4acnegative no real roots

44. THE DISCRIMINANT – MAKING PREDICTIONS Use the discriminant to predict how many “roots” each equation will have. 1. x2 – 7x – 2 = 0 49–4(1)(-2)=57 2 irrational roots 2. 0 = 2x2– 3x + 1 9–4(2)(1)=1  2 rational roots 3. 0 = 5x2 – 2x + 3 4–4(5)(3)=-56  no real roots 100–4(1)(25)=0  1 rational double root 4. x2– 10x + 25=0

45. THE DISCRIMINANT – MAKING PREDICTIONS about Parabolas The “zeros” of a function are the x-intercepts on it’s graph. Use the discriminant to predict how many x-intercepts each parabola will have and where the vertex is located. 1. y = 2x2 – x - 6 1–4(2)(-6)=49  2 rational zeros opens up/vertex below x-axis/2 x-intercepts 2. f(x) = 2x2 – x + 6 1–4(2)(6)=-47  no real zeros opens up/vertex above x-axis/No x-intercepts 3. y = -2x2– 9x + 6 81–4(-2)(6)=129 2 irrational zeros opens down/vertex above x-axis/2 x-intercepts 4. f(x) = x2 – 6x + 9 36–4(1)(9)=0  one rational zero opens up/vertex ON the x-axis/1 x-intercept

46. THE DISCRIMINANT – MAKING PREDICTIONS Note the proper terminology: The “zeros” of a function are the x-intercepts on it’s graph. Use the discriminant to predict how many x-intercepts each parabola will have. The “roots” of an equation are the x values that make the expression equal to zero. Equations have roots. Functions have zeros which are the x-intercepts on it’s graph.

47. PREVIEW/SUMMARY OF QUADRATIC EQUATIONS & FUNCTIONS END OF METHOD 4: QUADRATIC FORMULA Click here to return to menu slide

48. FOUR METHODS – HOW DO I CHOOSE? Some suggestions: Quadratic Formula – works for all quadratic equations, but look first for a “quicker” method. Don’t forget to simplify square roots and use value of discriminant to predict number of roots. Square Roots of Both Sides – use when the problem is easily written as glob2 = constant. Examples: 3(x + 2)2=12 or x2 – 75 = 0

49. FOUR METHODS – HOW DO I CHOOSE? Some suggestions: Factoring – doesn’t always work, but IF you see the factors, this is probably the quickest method. Examples: x2 – 8x = 0 has a GCF 4x2 – 12x + 9 = 0 is a PST x2 – x – 6 = 0 is easy to FOIL Complete the Square – best used when a = 1 and b is even (so you won’t need to use fractions). Examples: x2 – 6x + 1 = 0

50. PREVIEW/SUMMARY OF QUADRATIC EQUATIONS & FUNCTIONS END OF HOW TO CHOOSE A QUADRATIC METHOD Click here to return to menu slide