lyzzy7 2 years ago Which number makes this equation true? v^2 + 10v + 16 = (v + 8)(v + ?)

1. lyzzy7

@jim_thompson5910

2. ParthKohli

I know a shortcut... can I say it?

3. jim_thompson5910

if you were to set up a table, the lower right corner would have the value of 8*? where the answer would replace the question mark

4. lyzzy7

5. jim_thompson5910

8*? has to equal 16 so basically 8p = 16 p = ???

6. jim_thompson5910

7. lyzzy7

2

8. jim_thompson5910

yep, you nailed it

9. lyzzy7

yayyy :)

10. jim_thompson5910

so v^2 + 10v + 16 = (v + 8)(v + ?) turns into v^2 + 10v + 16 = (v + 8)(v + 2)

11. lyzzy7

What is the factored form of x^2 + 6x + 8?

12. ParthKohli

The shortcut says that in $$v^2 + \color{blue}{10}v + \rm something$$ and $$(v + \color{green}{a})(v + \color{green}{b})$$, the sum of the greens is the blue guy.

13. jim_thompson5910

find two numbers that a) multiply to 8 (last term) AND b) add to 6 (middle coefficient)

14. lyzzy7

im confused @ParthKohli

15. lyzzy7

2 and 4 . @jim_thompson5910

16. ParthKohli

It's simple. You have $$(v + 8)(v + ?)$$. This shortcut says that $$8 + ? = 10$$.

17. jim_thompson5910

So x^2 + 6x + 8 factors to (x+2)(x+4) The rule is that x^2 + bx + c factors to (x+m)(x+n) where m+n = b and m*n = c

18. lyzzy7

ohhh! @ParthKohli thanks :)

19. jim_thompson5910

So it's just a matter of finding those two numbers then writing it in the form (x+number1)(x+number2)

20. ParthKohli

I know that was sarcastic. -_-

21. lyzzy7

hold on im thinking @jim_thompson5910

22. lyzzy7

5 and 3?? @jim_thompson5910

23. jim_thompson5910

you found the numbers already by saying 2 and 4 2+4 = 6 (middle coefficient) 2*4 = 8 (last term)

24. lyzzy7

ohh.

25. jim_thompson5910

so because you found two numbers that multiply to 8 AND add to 6 the factorization of x^2 + 6x + 8 would be (x+2)(x+4)

26. lyzzy7

so it's (x + 4)(x + 2) ?

27. jim_thompson5910

go ahead and expand out (x+2)(x+4) you'll find that you'll get x^2 + 6x + 8 back again

28. jim_thompson5910

so essentially what you've done is gone in reverse of expanding (which was done in the previous problem)

29. lyzzy7

i mostly get what you're saying. i have one more if you're willing to help?

30. jim_thompson5910

correct, the order of the factors doesn't matter so (x+2)(x+4) is the same as (x+2)(x+4)

31. jim_thompson5910

i meant (x+2)(x+4) is the same as (x+4)(x+2)

32. jim_thompson5910

sure go for it

33. lyzzy7

What is the factored form of x^2 – 7x + 12? and dont worry i knew what you meant lol

34. jim_thompson5910

same idea, just different numbers

35. jim_thompson5910

find two numbers that multiply to 12 AND add to -7

36. lyzzy7

-3 and -4

37. jim_thompson5910

perfect

38. jim_thompson5910

so it factors to (x-3)(x-4)

39. lyzzy7

(x – 4)(x – 3) ?

40. jim_thompson5910

unfortunately this trick only works if the leading coefficient is 1, but it's still useful

41. jim_thompson5910

you got it

42. jim_thompson5910

that's the same as (x-3)(x-4) because the order of the factors does not matter ex: 2 time 3 = 6 and 3 times 2 = 6

43. lyzzy7

okay i have a different kind of problem... The area of a garden is given by the trinomial g2 – 2g – 24. The garden’s length is g + 4. What is the garden’s width?

44. jim_thompson5910

well it's actually the same type of problem, just worded differently

45. jim_thompson5910

they want you to factor g^2 - 2g - 24

46. jim_thompson5910

what do you get when you do?

47. lyzzy7

write me an equation then and let me solve it :)

48. lyzzy7

like for the question

49. jim_thompson5910

find me two numbers that multiply to -24 and add to -2 at the same time

50. lyzzy7

-6 and -4

51. jim_thompson5910

-6 plus -4 = -10 and it does NOT equal -2

52. jim_thompson5910

so try again

53. lyzzy7

oops lol hold on

54. lyzzy7

-6 and 4? -6*4= 24 -6+4 = -2 i think lol

55. jim_thompson5910

good

56. jim_thompson5910

So g^2 - 2g - 24 factors to (g-6)(g+4)

57. jim_thompson5910

notice how g+4 is a factor

58. jim_thompson5910

the other factor is g-6

59. jim_thompson5910

let's go back to the question The area of a garden is given by the trinomial g2 – 2g – 24. The garden’s length is g + 4. What is the garden’s width?

60. lyzzy7

g-6!

61. jim_thompson5910

correct since Area = Length times Width

62. lyzzy7

:D

63. jim_thompson5910

so if you have some area expression and you factor it, you get (Length)*(Width) where length and width are the factors of the area

64. lyzzy7

I actually get that!

65. jim_thompson5910

it only works for rectangles though

66. jim_thompson5910

that's great

67. lyzzy7

hb this one? What is the factored form of 4x^2 + 12x + 5?

68. jim_thompson5910

ok now the leading coefficient isn't 1 anymore, so we can't use that trick (well not the whole thing anyways)

69. jim_thompson5910

but we can still use some of the trick

70. jim_thompson5910

first multiply 4 (first coefficient) and 5 (last term) to get 4*5 = 20

71. jim_thompson5910

now list two numbers that multiply to 20 AND add to 12 (middle coefficient)

72. lyzzy7

2 and 10

73. lyzzy7

is it (4x + 5)(x + 1) ?

74. jim_thompson5910

so we break 12x into 2x + 10x then we factor by grouping 4x^2+12x+5 4x^2+2x+10x+5 (4x^2+2x)+(10x+5) 2x(2x+1)+(10x+5) 2x(2x+1)+5(2x+1) (2x+5)(2x+1)

75. jim_thompson5910

hopefully factoring by grouping is a familiar term

76. lyzzy7

so the final answer is (2x + 1)(2x + 5), right?

77. jim_thompson5910

correct

78. jim_thompson5910

hopefully you see how I got all that

79. lyzzy7

okay, thanks :D and lol yeah mostly.

80. lyzzy7

The area of a rectangular swimming pool is 10x2 – 19x – 15. The length of the pool is 5x + 3. What is the width of the pool?

81. jim_thompson5910

alright, just keep practicing and it'll make more sense

82. jim_thompson5910

10*(-15) = -150

83. jim_thompson5910

find two numbers that multiply to -150 and they add to -19

84. lyzzy7

why do you multiply 10 and -15?

85. jim_thompson5910

because I'm using the AC method for factoring this method starts out by multiplying the first and last coefficients

86. jim_thompson5910

then you find two numbers that multiply to that result and add to the middle number

87. lyzzy7

so the formula is ax^2 + bx + c right?

88. jim_thompson5910

that's a basic form of a quadratic, yes

89. jim_thompson5910

you would multiply 'a' and 'c' to get a*c or just ac then you find two numbers that multiply to ac and add to b

90. jim_thompson5910

so that's why they call it the ac method

91. lyzzy7

I cant find the two numbers that multiply to -150 and add to -19..

92. lyzzy7

im too dumb..

93. jim_thompson5910

no you're not

94. jim_thompson5910

stop that

95. jim_thompson5910

there's a long list to check, but luckily there's a shortcut

96. jim_thompson5910

have you heard of the quadratic formula before?

97. lyzzy7

noo. my teacher sucks this year. she is retiring so she doesnt teach us anything useful, she just assigns stuff and tells us to read our book to figure out how to do it, but the book has no tips or tricks.

98. jim_thompson5910

ah i'm sry to hear that, at least she'll be gone and a (hopefully) better teacher will take over

99. lyzzy7

yeah im going to a private school next year, sadly now i need to get a second job because its \$6,000 a year.

100. jim_thompson5910

dang that's quite pricey

101. jim_thompson5910

well anyways, the quadratic formula isn't that bad to learn (it's a bit ugly the first time you see it, but once you get used to it, it's not so bad)

102. lyzzy7

i know right, and okay shoot

103. jim_thompson5910

the quadratic formula is $\Large x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$ does that formula look familiar?

104. lyzzy7

noo.......

105. jim_thompson5910

ok but does it look readable by that I mean, could you simplify that formula if you had to (or if you had to plug in numbers for a, b, and c)?

106. lyzzy7

i guess so.

107. jim_thompson5910

ok in this case, a = 10, b = -19 and c = -15 we plug those values into the formula to get this (there's a lot of steps, but just take things one step at a time) $\Large x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$ $\Large x = \frac{-(-19)\pm\sqrt{(-19)^2-4(10)(-15)}}{2(10)}$ $\Large x = \frac{19\pm\sqrt{361-(-600)}}{20}$ $\Large x = \frac{19\pm\sqrt{361+600}}{20}$ $\Large x = \frac{19\pm\sqrt{961}}{20}$ $\Large x = \frac{19+\sqrt{961}}{20} \ \text{or} \ x = \frac{19-\sqrt{961}}{20}$ $\Large x = \frac{19+31}{20} \ \text{or} \ x = \frac{19-31}{20}$ $\Large x = \frac{50}{20} \ \text{or} \ x = \frac{-12}{20}$ $\Large x = \frac{5}{2} \ \text{or} \ x = -\frac{3}{5}$ Let me know when you want to continue.

108. lyzzy7

I honestly have no clue what any of that is. att all. like at all. im taking algebra 1 btw. im in 8th grade. but i really have cant even read that...

109. lyzzy7

like i cant even read it in my head let alone get it. i have a headache.

110. lyzzy7

fml. i hate math.

111. jim_thompson5910

alright, sorry jumped too far ahead

112. lyzzy7

its not your fault. it just is ridiculous that something that complicated even exists. i hate this world.

113. jim_thompson5910

well much more complicated things exist (that even I don't understand), but that's what keeps things interesting

114. jim_thompson5910

let's ignore the quadratic formula until you get more practice with it

115. lyzzy7

soo what is the answer to this? The area of a rectangular swimming pool is 10x2 – 19x – 15. The length of the pool is 5x + 3. What is the width of the pool? its like 5 am where i live and i still havent slept.

116. jim_thompson5910

why not sleep then come back?

117. jim_thompson5910

10(-15) = -150 find two numbers that multiply to -150 and add to -19 ....do a bunch of trial and error guesses... you'll find that the two numbers are 6 and -25 So this means... 10x^2-19x-15 10x^2+6x-25x-15 (10x^2+6x)+(-25x-15) 2x(5x+3)+(-25x-15) 2x(5x+3)-5(5x+3) (2x-5)(5x+3) --------------------------------------------------------------------------- In short 10x^2-19x-15 completely factors to (2x-5)(5x+3)

118. lyzzy7

because the kind of person i am, when i have something on my mind, i have to get it done before i'd ever care for myself.

119. jim_thompson5910

i gotcha, that's both a good and bad trait because yes it's good to be this motivated, but not when you're sleep deprived (you won't learn much without a good night's sleep)

120. jim_thompson5910

so anyways, back to the problem above I showed how 10x^2-19x-15 factors to (2x-5)(5x+3) 5x+3 was one factor we were already given, 2x-5 is the other one So the width is 2x-5

121. lyzzy7

Got it. and the length is 5x+3 right?

122. jim_thompson5910

so length = 5x + 3 (given) width = 2x - 5 (just found this)

123. jim_thompson5910

correct

124. lyzzy7

ughhh now i have to solve two more, then ima sleep. thank you so much for the help!

125. jim_thompson5910

sure thing

126. jim_thompson5910

go ahead and post and hopefully it won't take too long

127. jim_thompson5910

then you can finally sleep (til you have to get up in the morning...ugh lol)

128. lyzzy7

you sure? i mean i could try to do them on my own if you want.

129. jim_thompson5910

sure whatever works best for you

130. jim_thompson5910

how about you work them and I check what you get

131. jim_thompson5910

or I can show you how if you're stuck

132. lyzzy7

alright heres this one What is the factored form of 2x2 + x – 3? then i'll do the other.

133. jim_thompson5910

ok 2(3) = -6 find two numbers that multiply to -6 and add to 1 those two numbers are 3 and -2 so... 2x^2+x-3 2x^2+3x-2x-3 ... break 1x into 3x - 2x, then factor by grouping (2x^2+3x)+(-2x-3) x(2x+3)+(-2x-3) x(2x+3)-1(2x+3) (x-1)(2x+3) ------------------------------------- 2x^2+x-3 completely factors to (x-1)(2x+3)

134. jim_thompson5910

hopefully I didn't cut you off there?

135. lyzzy7

no you're good, i was just going to say the two numbers. but how in god's name did you do that so fast....

136. jim_thompson5910

lol magic

137. jim_thompson5910

ah i should have let you say the two numbers, but hopefully you got 3 and -2 ?

138. lyzzy7

yeah i did lol

139. jim_thompson5910

ok perfect, you'll be a pro in no time

140. jim_thompson5910

you just need to practice more with factoring by grouping (and the quadratic formula when you get more time)

141. lyzzy7

and for the other one it was What is the factored form of 3x2 + 21x – 24? and i got 3(x + 8)(x – 1) am i almost right?

142. jim_thompson5910

no, you're not almost right....you're absolutely correct (lol I wanted to sound like a game show host for a sec)

143. jim_thompson5910

nice work

144. lyzzy7

OMG YAY:))) Thank you sooo much! like i really didnt get half my stuff till you helped lol

145. jim_thompson5910

glad to be of help, now get some sleep lol

146. lyzzy7

will do, thank you :)

147. jim_thompson5910

I know, I probably sound like your mom, but idc lol

148. jim_thompson5910

yw

149. lyzzy7

your good lol! how old are u by the way?

150. lyzzy7

youre*

151. lyzzy7

lol nah its cool. and its amazing that you put your brain to help people. what do you do as a job?

152. jim_thompson5910

but thanks lol

153. lyzzy7

i hope you go far! honestly

154. jim_thompson5910

thanks, i hope so too

155. jim_thompson5910

definitely getting a lot of practice here lol

156. lyzzy7

lol, there ya go!

157. jim_thompson5910

and building networks as well (well sorta i guess)

158. jim_thompson5910

so that's another plus