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

- anonymous

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

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- chestercat

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- anonymous

@jim_thompson5910

- ParthKohli

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

- 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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## More answers

- anonymous

yes please @ParthKohli

- jim_thompson5910

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

- jim_thompson5910

where p is that answer

- anonymous

2

- jim_thompson5910

yep, you nailed it

- anonymous

yayyy :)

- jim_thompson5910

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

- anonymous

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

- 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

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

- anonymous

im confused @ParthKohli

- anonymous

2 and 4 . @jim_thompson5910

- ParthKohli

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

- 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

ohhh! @ParthKohli thanks :)

- jim_thompson5910

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

- ParthKohli

I know that was sarcastic. -_-

- anonymous

hold on im thinking @jim_thompson5910

- anonymous

5 and 3?? @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

ohh.

- 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

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

- 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

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

- anonymous

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

- 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

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

- jim_thompson5910

sure go for it

- anonymous

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

- jim_thompson5910

same idea, just different numbers

- jim_thompson5910

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

- anonymous

-3 and -4

- jim_thompson5910

perfect

- jim_thompson5910

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

- anonymous

(x – 4)(x – 3) ?

- jim_thompson5910

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

- jim_thompson5910

you got it

- 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

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

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

- jim_thompson5910

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

- jim_thompson5910

what do you get when you do?

- anonymous

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

- anonymous

like for the question

- jim_thompson5910

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

- anonymous

-6 and -4

- jim_thompson5910

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

- jim_thompson5910

so try again

- anonymous

oops lol hold on

- anonymous

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

- jim_thompson5910

good

- jim_thompson5910

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

- jim_thompson5910

notice how g+4 is a factor

- jim_thompson5910

the other factor is g-6

- 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

g-6!

- jim_thompson5910

correct since Area = Length times Width

- anonymous

:D

- 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

I actually get that!

- jim_thompson5910

it only works for rectangles though

- jim_thompson5910

that's great

- anonymous

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

- 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

but we can still use some of the trick

- jim_thompson5910

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

- jim_thompson5910

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

- anonymous

2 and 10

- anonymous

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

- 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

hopefully factoring by grouping is a familiar term

- anonymous

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

- jim_thompson5910

correct

- jim_thompson5910

hopefully you see how I got all that

- anonymous

okay, thanks :D and lol yeah mostly.

- 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

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

- jim_thompson5910

10*(-15) = -150

- jim_thompson5910

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

- anonymous

why do you multiply 10 and -15?

- jim_thompson5910

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

- jim_thompson5910

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

- anonymous

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

- jim_thompson5910

that's a basic form of a quadratic, yes

- 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

so that's why they call it the ac method

- anonymous

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

- anonymous

im too dumb..

- jim_thompson5910

no you're not

- jim_thompson5910

stop that

- jim_thompson5910

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

- jim_thompson5910

have you heard of the quadratic formula before?

- 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

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

- 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

dang that's quite pricey

- 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

i know right, and okay shoot

- jim_thompson5910

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

- anonymous

noo.......

- 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

i guess so.

- 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

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

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

- anonymous

fml. i hate math.

- jim_thompson5910

alright, sorry jumped too far ahead

- anonymous

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

- jim_thompson5910

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

- jim_thompson5910

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

- 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

why not sleep then come back?

- 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

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

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

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

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

- jim_thompson5910

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

- jim_thompson5910

correct

- anonymous

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

- jim_thompson5910

sure thing

- jim_thompson5910

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

- jim_thompson5910

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

- anonymous

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

- jim_thompson5910

sure whatever works best for you

- jim_thompson5910

how about you work them and I check what you get

- jim_thompson5910

or I can show you how if you're stuck

- anonymous

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

- 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

hopefully I didn't cut you off there?

- 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

lol magic

- jim_thompson5910

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

- anonymous

yeah i did lol

- jim_thompson5910

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

- jim_thompson5910

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

- 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

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

nice work

- anonymous

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

- jim_thompson5910

glad to be of help, now get some sleep lol

- anonymous

will do, thank you :)

- jim_thompson5910

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

- jim_thompson5910

yw

- anonymous

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

- anonymous

youre*

- 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

but thanks lol

- anonymous

i hope you go far! honestly

- jim_thompson5910

thanks, i hope so too

- jim_thompson5910

definitely getting a lot of practice here lol

- anonymous

lol, there ya go!

- jim_thompson5910

and building networks as well (well sorta i guess)

- jim_thompson5910

so that's another plus

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