7D Factoring and Solving Quadratic Equations y = x2 + 8x is equivalent to y-intercept, AOS, vertex y = 2x2 + 9x + 4 y = x(x + 8) x-intercepts (solve x=0) y = (2x + 1)(x + 4) FACTORING Greatest Common Factor (any number of terms) To go from standard form to x-intercept form, you must factor Method 1: Factor by GCF (any number of terms) GCF: Largest number that goes into everything Smallest exponent of each variable Ex 1: 15x2y3, 10x4y, 20x3y2z GCF: 5x2y because x2,y1,zo are smallest exponent To factor by GCF: 1. Find the GCF and put it on the “outside” of ( ) 2. Divide everything by the GCF and put result inside ( Ex 2: Factor 10x2 + 20x 2 GCF is 10x so 10 x 20 x = x + 2 Answer 10x(x + 2) 10 x 10 x Check answer: 10x(x + 2) = 10x2 + 20x Ex 3: Solve 12x2 – 6x = 0 2 GCF is 6x so 12 x 6 x = 2x - 1 6x(2x -1) = 0 6x 6x 6x = 0 2x+ 1 = 0 x = 0 2x = -1 x=-1/2 2 5x2- 15x = 5 x 15 = 5x(1-3x) 5x ) Difference of Squares (2) (a2 – c2) = (a-c)(a+c) Or ax2 + 0x – c X2 – 16 = (x-4)(x+4) 49x2 – 1 =(7x2+1)(7x2-1) 5x Grouping (4 terms) 5x2 – 10x – 2x + 4 5x (x – 2) – 2(x - 2) (5x – 2) (x – 2) Trinomial x2 + bx + c (3 terms) (x +bx +c ) = (+) (+) (x2 -bx +c ) =(-)(-) 2 ( x ?bx -c ) = ( - )( + ) or (+)(-) 2 mn=c m+n = b x2 + mx + nx + c Solve by grouping X2 – 6x – 40 ( - )( +) m=-10, n=4 -10*4=-40, -10+4=-6 X2 – 10x + 4x – 40 X(x – 10) + 4(x – 10) (X + 4)(x – 10) Method 2: Grouping Method (4 terms) 1. Factor the first two terms by GCF 2. Factor the last two terms by GCF making sure that the second ( ) is same as the first ( ) 3. Rewrite in a similar manner as: x(a + b) + c(a + b) (x + c)(a + b) Ex 4: Factor x3 + 2x2 + 4x + 8 Ex 5: Factor 2x3 – 4x2 – x + 2 2 3 2 GCF is x (x + 2x ) + (4x + 8) GCF is 4x GCF is 2x2 (2x3 + (– 4x2) + (-x + 2) GCF is -1 or +1 3 2 x 2x 4x 8 2 x3 4 x 2 2x2(x -2) + -1(x – 2) Divide by -1 to get (x-2) 2 2 x (x + 2) + 4x(x + 2) 2 2 x2 2 x2 x x 4 4 (x2 + 4x)(x + 2) (2x2- 1)(x – 2) Sung to “If you’re happy and you know it…” Method 3: Solve by Making Groups (3 terms) x2 + bx + c ( ax2 + bx + c ) 1. Determine the “signs” of the ( )( )If c is ‘+’ then 2 of the same signs as ‘b’ FIRST SECOND If the second is a plus, two of the first ( + +) = (+)(+) If c is ‘-‘ then 1 ‘+’ and 1 ‘-‘ If the second is a plus, two of the first ( - +) = (- )(-) 2. Find two numbers m,n so than mn =c and m+n = b If the second is a plus, then you add to get the middle, If c is ‘+’ then m,n will be same sign as ‘b.’ If the second is a plus, two of the first If c is ‘-‘ then m,n will be opposite signs. 3. Split bx into mx + nx If the second is a minus, one of each ( ? - ) = (+)(-) or (-)(+) If the second is a minus, one of each 4. Factor by grouping If the second is a minus, then you subtract to get the Ex 6: x2 + 6x + 8 Ex 7: x2 – 9x - 10 middle 1. Signs are ( + )( + ) 1. Signs are ( - )( +) or (+)(-) If the second is a minus, one of each 2. 8 is 1*8 or 2*4. Use 2*4 since 2+4 = 6 2. -10 is 2*(-5), -2*5,(-10)*1, 10*(-1) Use -10*1 since it adds to -9 3. (x2 + 2x) + (4x + 8) 3. (x2 – 10x) + (1x – 10) 4. x(x+2) + 4(x+2) 4. x(x – 10) + 1(x – 10) (x+4)(x+2) (x + 1)(x – 10) Ex 8: Solve 1. 2. 3. 4. x2 – 10x + 24 = 0 Ex 9: Signs are ( - ) ( - ) 24 = (-1)(-12), (-2)(-12), (-3)(-8), (-4)(-6) (x2 – 4x) + (-6x + 24) = 0 x(x – 4) + -6(x – 4) = 0 (x – 6)(x – 4) = 0 x-6= 0 x–4=0 x=6 x=4 x2 + 10x = 24 x2 + 10x – 24 = 0 (Must be = 0) 1. Signs are ( - )(+) or (+)(-) 2. -24 = (-1)(24), (-2)(12), (2)(-12)etc..Use 2&-12 since 2+-12 = -10 3. (x2 – 12x) + (2x – 24) = 0 4. x(x – 12) + 2( x – 12) = 0 (x+2)(x -12) = 0 x+ 2 = 0 x – 12 = 0 x = -2 x = 12 Method 4: Solve by difference of Squares (2 terms) x2 – c 1. Rewrite equation as x2 + 0x – c or remember m2 – n2 = (m-n)(m+n) 2. Factor using Method 3 (signs will always be (-)(+) Ex 10: x2 – 9 Ex 11: x2 = 81 2 1. x + 0x – 9 or (x-3)(x+3) x2 – 81 = 0 2. -9 = (-9)(1) or (-3)(3). Use -3,3 since -3+3=0 x2 + 0x – 81=0 2 x – 3x + 3x – 9 x2 – 9x + 9x – 81=0 x(x-3)+3(x-3) x(x-9) + 9(x – 9)=0 (x+3)(x-3) (x + 9)(x – 9) = 0 x+9=0 x–9=0 x = -9 x=9 Combination Method Take out GCF and then use Methods 2,3 and/or 4 Ex 12: -x2 + 5x + 6 = 0 Ex 13: 6x3 – 54x = 0 Take the GCF of -1 out Take GCF of 6x out 6x(x2 – 9) Use difference of squares (method 4) 2 -1(x – 5x – 6) = 0 (-)(+) -6=(-2)(3), (6)(-1), (-6)(1) 6x(x2 + 0x – 9) = 0 2 -1(x – 6x + x – 6)=0 6x(x2 + 3x – 3x – 9) = 0 -1(x(x-6) + 1(x-6)) = 0 6x(x(x+3) – 3(x +3))=0 -1(x+1)(x-6) = 0 6x(x-3)(x+3) = 0 -1= 0 x+1 = 0 x-6 = 0 6x = 0 x-3 = 0 x+3 = 0 x = -1 x=6 x = 0 x=3 x=-3 Practice 1-5 Find the GCF 1. 5x2 , 25x 2. 12x3,18x2 6-10 Rewrite in factored form using GCF Method 6. y = 10x2 + 2x 7. y = 4x2 + 16x 11-15 Find the x-intercepts 11. y = 15x2 – 20x 14. y = -2x2 + 12x 3. 10xz2,24x2 4. 14x3,12x2,20x4 5. 24x3y4z3, 18x2yz5 8. y = x2 + 7x 9. y = 10x2 – 5x 10. y = 2x2 – 12x (Set = 0, factor and then solve) 12. y = 12x2 – 8x 13. y = -x2 – 8x 15. y = 3x2 – 12x 16-19 Rewrite in factored form using the Grouping Method 16. 2xy + 7x – 2y – 7 17. 15a – 3ab - 4b + 20. 18. x2 + 12x + 3x + 36 19. 5x2 – 15x – 3x + 9 20-28 Rewrite in factored form using the trinomial method (#3) 20. a2 + 8a + 15 21. c2 + 12c + 35 22. m2 – 22m + 21 23. p2 – 17p + 72 24. x2 + 6x – 7 26. y2 – y – 42 25. h2 + 3h – 40 27. -72 + 6w + w2 (hint rewrite the order) 28. a2 + 5ab + 4b2 29-33. Solve for x. (Rewrite each side = 0) 29. x2 – 5x + 6 = 0 30. x2 – 3x = 18 32. x2 – 7x – 6=2 33. x2 – 8x + 48 = 0 34-38 Factor using the Difference of Squares Method (#4) 34. x2 – 16 35. x2 – 100 36. 4x2 – 25 37. 9x2 – 64 38. x4 – 49 39-43 Solve using the Difference of Squares Method (#4) 39. x2 – 1 = 0 40. x2 – 36 = 0 41. x2 = 25 42. x2 = 64 31. x2 = 5x + 6 44-48 Solve using GCF and then continue factoring. 44. –x2 + 6x + 7 = 0 45. –x2 + 10x + 24 = 0 47. 2x2 – 4x – 16 = 0 43. 25x2 – 9 = 0 46. 4x2 – 36 = 0 48. 3x2 = 12x 49-52 Choose the correct method and then factor 49. x2 – 36 50. x3 + 3x2 + 6x + 18 51. 16x2y – 56xy2 – 8x2 52. x2 – 8x – 20 53- 55 Find a possible integral length and width of a rectangle with the following area: (Hint just factor) 53. x2 + 12x + 20 54. x2 + 6x – 16 55. x2 – 16 Solve the following equations 56. x2 = 8x 59. x2 + 8x = 9 57. x2 = 100 58. –x2 – 10x – 24 = 0 60. x2 – 6x = 2x + 33 61. The height of a ball is given by the equation h(t) = -t2 + 4t + 5. Find where it hits the ground (Hint set equation = 0)