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)