anonymous
  • anonymous
Which number makes this equation true? v^2 + 10v + 16 = (v + 8)(v + ?)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
anonymous
  • anonymous
@jim_thompson5910
ParthKohli
  • ParthKohli
I know a shortcut... can I say it?
jim_thompson5910
  • 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

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anonymous
  • anonymous
yes please @ParthKohli
jim_thompson5910
  • jim_thompson5910
8*? has to equal 16 so basically 8p = 16 p = ???
jim_thompson5910
  • jim_thompson5910
where p is that answer
anonymous
  • anonymous
2
jim_thompson5910
  • jim_thompson5910
yep, you nailed it
anonymous
  • anonymous
yayyy :)
jim_thompson5910
  • jim_thompson5910
so v^2 + 10v + 16 = (v + 8)(v + ?) turns into v^2 + 10v + 16 = (v + 8)(v + 2)
anonymous
  • anonymous
What is the factored form of x^2 + 6x + 8?
ParthKohli
  • 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.
jim_thompson5910
  • jim_thompson5910
find two numbers that a) multiply to 8 (last term) AND b) add to 6 (middle coefficient)
anonymous
  • anonymous
im confused @ParthKohli
anonymous
  • anonymous
2 and 4 . @jim_thompson5910
ParthKohli
  • ParthKohli
It's simple. You have \((v + 8)(v + ?)\). This shortcut says that \(8 + ? = 10\).
jim_thompson5910
  • 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
anonymous
  • anonymous
ohhh! @ParthKohli thanks :)
jim_thompson5910
  • jim_thompson5910
So it's just a matter of finding those two numbers then writing it in the form (x+number1)(x+number2)
ParthKohli
  • ParthKohli
I know that was sarcastic. -_-
anonymous
  • anonymous
hold on im thinking @jim_thompson5910
anonymous
  • anonymous
5 and 3?? @jim_thompson5910
jim_thompson5910
  • jim_thompson5910
you found the numbers already by saying 2 and 4 2+4 = 6 (middle coefficient) 2*4 = 8 (last term)
anonymous
  • anonymous
ohh.
jim_thompson5910
  • 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)
anonymous
  • anonymous
so it's (x + 4)(x + 2) ?
jim_thompson5910
  • jim_thompson5910
go ahead and expand out (x+2)(x+4) you'll find that you'll get x^2 + 6x + 8 back again
jim_thompson5910
  • jim_thompson5910
so essentially what you've done is gone in reverse of expanding (which was done in the previous problem)
anonymous
  • anonymous
i mostly get what you're saying. i have one more if you're willing to help?
jim_thompson5910
  • jim_thompson5910
correct, the order of the factors doesn't matter so (x+2)(x+4) is the same as (x+2)(x+4)
jim_thompson5910
  • jim_thompson5910
i meant (x+2)(x+4) is the same as (x+4)(x+2)
jim_thompson5910
  • jim_thompson5910
sure go for it
anonymous
  • anonymous
What is the factored form of x^2 – 7x + 12? and dont worry i knew what you meant lol
jim_thompson5910
  • jim_thompson5910
same idea, just different numbers
jim_thompson5910
  • jim_thompson5910
find two numbers that multiply to 12 AND add to -7
anonymous
  • anonymous
-3 and -4
jim_thompson5910
  • jim_thompson5910
perfect
jim_thompson5910
  • jim_thompson5910
so it factors to (x-3)(x-4)
anonymous
  • anonymous
(x – 4)(x – 3) ?
jim_thompson5910
  • jim_thompson5910
unfortunately this trick only works if the leading coefficient is 1, but it's still useful
jim_thompson5910
  • jim_thompson5910
you got it
jim_thompson5910
  • 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
anonymous
  • anonymous
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?
jim_thompson5910
  • jim_thompson5910
well it's actually the same type of problem, just worded differently
jim_thompson5910
  • jim_thompson5910
they want you to factor g^2 - 2g - 24
jim_thompson5910
  • jim_thompson5910
what do you get when you do?
anonymous
  • anonymous
write me an equation then and let me solve it :)
anonymous
  • anonymous
like for the question
jim_thompson5910
  • jim_thompson5910
find me two numbers that multiply to -24 and add to -2 at the same time
anonymous
  • anonymous
-6 and -4
jim_thompson5910
  • jim_thompson5910
-6 plus -4 = -10 and it does NOT equal -2
jim_thompson5910
  • jim_thompson5910
so try again
anonymous
  • anonymous
oops lol hold on
anonymous
  • anonymous
-6 and 4? -6*4= 24 -6+4 = -2 i think lol
jim_thompson5910
  • jim_thompson5910
good
jim_thompson5910
  • jim_thompson5910
So g^2 - 2g - 24 factors to (g-6)(g+4)
jim_thompson5910
  • jim_thompson5910
notice how g+4 is a factor
jim_thompson5910
  • jim_thompson5910
the other factor is g-6
jim_thompson5910
  • 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?
anonymous
  • anonymous
g-6!
jim_thompson5910
  • jim_thompson5910
correct since Area = Length times Width
anonymous
  • anonymous
:D
jim_thompson5910
  • 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
anonymous
  • anonymous
I actually get that!
jim_thompson5910
  • jim_thompson5910
it only works for rectangles though
jim_thompson5910
  • jim_thompson5910
that's great
anonymous
  • anonymous
hb this one? What is the factored form of 4x^2 + 12x + 5?
jim_thompson5910
  • jim_thompson5910
ok now the leading coefficient isn't 1 anymore, so we can't use that trick (well not the whole thing anyways)
jim_thompson5910
  • jim_thompson5910
but we can still use some of the trick
jim_thompson5910
  • jim_thompson5910
first multiply 4 (first coefficient) and 5 (last term) to get 4*5 = 20
jim_thompson5910
  • jim_thompson5910
now list two numbers that multiply to 20 AND add to 12 (middle coefficient)
anonymous
  • anonymous
2 and 10
anonymous
  • anonymous
is it (4x + 5)(x + 1) ?
jim_thompson5910
  • 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)
jim_thompson5910
  • jim_thompson5910
hopefully factoring by grouping is a familiar term
anonymous
  • anonymous
so the final answer is (2x + 1)(2x + 5), right?
jim_thompson5910
  • jim_thompson5910
correct
jim_thompson5910
  • jim_thompson5910
hopefully you see how I got all that
anonymous
  • anonymous
okay, thanks :D and lol yeah mostly.
anonymous
  • anonymous
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?
jim_thompson5910
  • jim_thompson5910
alright, just keep practicing and it'll make more sense
jim_thompson5910
  • jim_thompson5910
10*(-15) = -150
jim_thompson5910
  • jim_thompson5910
find two numbers that multiply to -150 and they add to -19
anonymous
  • anonymous
why do you multiply 10 and -15?
jim_thompson5910
  • jim_thompson5910
because I'm using the AC method for factoring this method starts out by multiplying the first and last coefficients
jim_thompson5910
  • jim_thompson5910
then you find two numbers that multiply to that result and add to the middle number
anonymous
  • anonymous
so the formula is ax^2 + bx + c right?
jim_thompson5910
  • jim_thompson5910
that's a basic form of a quadratic, yes
jim_thompson5910
  • 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
jim_thompson5910
  • jim_thompson5910
so that's why they call it the ac method
anonymous
  • anonymous
I cant find the two numbers that multiply to -150 and add to -19..
anonymous
  • anonymous
im too dumb..
jim_thompson5910
  • jim_thompson5910
no you're not
jim_thompson5910
  • jim_thompson5910
stop that
jim_thompson5910
  • jim_thompson5910
there's a long list to check, but luckily there's a shortcut
jim_thompson5910
  • jim_thompson5910
have you heard of the quadratic formula before?
anonymous
  • anonymous
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.
jim_thompson5910
  • jim_thompson5910
ah i'm sry to hear that, at least she'll be gone and a (hopefully) better teacher will take over
anonymous
  • anonymous
yeah im going to a private school next year, sadly now i need to get a second job because its $6,000 a year.
jim_thompson5910
  • jim_thompson5910
dang that's quite pricey
jim_thompson5910
  • 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)
anonymous
  • anonymous
i know right, and okay shoot
jim_thompson5910
  • jim_thompson5910
the quadratic formula is \[\Large x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\] does that formula look familiar?
anonymous
  • anonymous
noo.......
jim_thompson5910
  • 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)?
anonymous
  • anonymous
i guess so.
jim_thompson5910
  • 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.
anonymous
  • anonymous
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...
anonymous
  • anonymous
like i cant even read it in my head let alone get it. i have a headache.
anonymous
  • anonymous
fml. i hate math.
jim_thompson5910
  • jim_thompson5910
alright, sorry jumped too far ahead
anonymous
  • anonymous
its not your fault. it just is ridiculous that something that complicated even exists. i hate this world.
jim_thompson5910
  • jim_thompson5910
well much more complicated things exist (that even I don't understand), but that's what keeps things interesting
jim_thompson5910
  • jim_thompson5910
let's ignore the quadratic formula until you get more practice with it
anonymous
  • anonymous
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.
jim_thompson5910
  • jim_thompson5910
why not sleep then come back?
jim_thompson5910
  • 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)
anonymous
  • anonymous
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.
jim_thompson5910
  • 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)
jim_thompson5910
  • 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
anonymous
  • anonymous
Got it. and the length is 5x+3 right?
jim_thompson5910
  • jim_thompson5910
so length = 5x + 3 (given) width = 2x - 5 (just found this)
jim_thompson5910
  • jim_thompson5910
correct
anonymous
  • anonymous
ughhh now i have to solve two more, then ima sleep. thank you so much for the help!
jim_thompson5910
  • jim_thompson5910
sure thing
jim_thompson5910
  • jim_thompson5910
go ahead and post and hopefully it won't take too long
jim_thompson5910
  • jim_thompson5910
then you can finally sleep (til you have to get up in the morning...ugh lol)
anonymous
  • anonymous
you sure? i mean i could try to do them on my own if you want.
jim_thompson5910
  • jim_thompson5910
sure whatever works best for you
jim_thompson5910
  • jim_thompson5910
how about you work them and I check what you get
jim_thompson5910
  • jim_thompson5910
or I can show you how if you're stuck
anonymous
  • anonymous
alright heres this one What is the factored form of 2x2 + x – 3? then i'll do the other.
jim_thompson5910
  • 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)
jim_thompson5910
  • jim_thompson5910
hopefully I didn't cut you off there?
anonymous
  • anonymous
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....
jim_thompson5910
  • jim_thompson5910
lol magic
jim_thompson5910
  • jim_thompson5910
ah i should have let you say the two numbers, but hopefully you got 3 and -2 ?
anonymous
  • anonymous
yeah i did lol
jim_thompson5910
  • jim_thompson5910
ok perfect, you'll be a pro in no time
jim_thompson5910
  • jim_thompson5910
you just need to practice more with factoring by grouping (and the quadratic formula when you get more time)
anonymous
  • anonymous
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?
jim_thompson5910
  • 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)
jim_thompson5910
  • jim_thompson5910
nice work
anonymous
  • anonymous
OMG YAY:))) Thank you sooo much! like i really didnt get half my stuff till you helped lol
jim_thompson5910
  • jim_thompson5910
glad to be of help, now get some sleep lol
anonymous
  • anonymous
will do, thank you :)
jim_thompson5910
  • jim_thompson5910
I know, I probably sound like your mom, but idc lol
jim_thompson5910
  • jim_thompson5910
yw
anonymous
  • anonymous
your good lol! how old are u by the way?
anonymous
  • anonymous
youre*
anonymous
  • anonymous
lol nah its cool. and its amazing that you put your brain to help people. what do you do as a job?
jim_thompson5910
  • jim_thompson5910
but thanks lol
anonymous
  • anonymous
i hope you go far! honestly
jim_thompson5910
  • jim_thompson5910
thanks, i hope so too
jim_thompson5910
  • jim_thompson5910
definitely getting a lot of practice here lol
anonymous
  • anonymous
lol, there ya go!
jim_thompson5910
  • jim_thompson5910
and building networks as well (well sorta i guess)
jim_thompson5910
  • jim_thompson5910
so that's another plus

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