lyzzy7
Which number makes this equation true?
v^2 + 10v + 16 = (v + 8)(v + ?)
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lyzzy7
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@jim_thompson5910
ParthKohli
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I know a shortcut... can I say it?
jim_thompson5910
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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
lyzzy7
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yes please @ParthKohli
jim_thompson5910
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8*? has to equal 16
so basically
8p = 16
p = ???
jim_thompson5910
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where p is that answer
lyzzy7
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jim_thompson5910
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yep, you nailed it
lyzzy7
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yayyy :)
jim_thompson5910
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so
v^2 + 10v + 16 = (v + 8)(v + ?)
turns into
v^2 + 10v + 16 = (v + 8)(v + 2)
lyzzy7
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What is the factored form of x^2 + 6x + 8?
ParthKohli
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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
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find two numbers that
a) multiply to 8 (last term)
AND
b) add to 6 (middle coefficient)
lyzzy7
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im confused @ParthKohli
lyzzy7
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2 and 4 . @jim_thompson5910
ParthKohli
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It's simple.
You have \((v + 8)(v + ?)\). This shortcut says that \(8 + ? = 10\).
jim_thompson5910
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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
lyzzy7
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ohhh! @ParthKohli thanks :)
jim_thompson5910
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So it's just a matter of finding those two numbers then writing it in the form (x+number1)(x+number2)
ParthKohli
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I know that was sarcastic. -_-
lyzzy7
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hold on im thinking @jim_thompson5910
lyzzy7
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5 and 3?? @jim_thompson5910
jim_thompson5910
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you found the numbers already by saying 2 and 4
2+4 = 6 (middle coefficient)
2*4 = 8 (last term)
lyzzy7
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ohh.
jim_thompson5910
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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)
lyzzy7
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so it's (x + 4)(x + 2) ?
jim_thompson5910
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go ahead and expand out (x+2)(x+4)
you'll find that you'll get x^2 + 6x + 8 back again
jim_thompson5910
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so essentially what you've done is gone in reverse of expanding (which was done in the previous problem)
lyzzy7
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i mostly get what you're saying. i have one more if you're willing to help?
jim_thompson5910
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correct, the order of the factors doesn't matter
so (x+2)(x+4) is the same as (x+2)(x+4)
jim_thompson5910
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i meant (x+2)(x+4) is the same as (x+4)(x+2)
jim_thompson5910
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sure go for it
lyzzy7
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What is the factored form of x^2 – 7x + 12?
and dont worry i knew what you meant lol
jim_thompson5910
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same idea, just different numbers
jim_thompson5910
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find two numbers that multiply to 12 AND add to -7
lyzzy7
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-3 and -4
jim_thompson5910
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perfect
jim_thompson5910
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so it factors to (x-3)(x-4)
lyzzy7
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(x – 4)(x – 3) ?
jim_thompson5910
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unfortunately this trick only works if the leading coefficient is 1, but it's still useful
jim_thompson5910
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you got it
jim_thompson5910
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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
lyzzy7
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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
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well it's actually the same type of problem, just worded differently
jim_thompson5910
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they want you to factor g^2 - 2g - 24
jim_thompson5910
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what do you get when you do?
lyzzy7
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write me an equation then and let me solve it :)
lyzzy7
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like for the question
jim_thompson5910
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find me two numbers that multiply to -24 and add to -2 at the same time
lyzzy7
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-6 and -4
jim_thompson5910
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-6 plus -4 = -10 and it does NOT equal -2
jim_thompson5910
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so try again
lyzzy7
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oops lol hold on
lyzzy7
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-6 and 4?
-6*4= 24
-6+4 = -2
i think lol
jim_thompson5910
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So g^2 - 2g - 24 factors to (g-6)(g+4)
jim_thompson5910
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notice how g+4 is a factor
jim_thompson5910
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the other factor is g-6
jim_thompson5910
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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?
lyzzy7
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g-6!
jim_thompson5910
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correct since Area = Length times Width
lyzzy7
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:D
jim_thompson5910
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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
lyzzy7
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I actually get that!
jim_thompson5910
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it only works for rectangles though
jim_thompson5910
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that's great
lyzzy7
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hb this one?
What is the factored form of 4x^2 + 12x + 5?
jim_thompson5910
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ok now the leading coefficient isn't 1 anymore, so we can't use that trick (well not the whole thing anyways)
jim_thompson5910
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but we can still use some of the trick
jim_thompson5910
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first multiply 4 (first coefficient) and 5 (last term) to get 4*5 = 20
jim_thompson5910
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now list two numbers that multiply to 20 AND add to 12 (middle coefficient)
lyzzy7
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2 and 10
lyzzy7
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is it (4x + 5)(x + 1) ?
jim_thompson5910
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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
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hopefully factoring by grouping is a familiar term
lyzzy7
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so the final answer is (2x + 1)(2x + 5), right?
jim_thompson5910
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correct
jim_thompson5910
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hopefully you see how I got all that
lyzzy7
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okay, thanks :D and lol yeah mostly.
lyzzy7
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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
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alright, just keep practicing and it'll make more sense
jim_thompson5910
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10*(-15) = -150
jim_thompson5910
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find two numbers that multiply to -150
and
they add to -19
lyzzy7
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why do you multiply 10 and -15?
jim_thompson5910
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because I'm using the AC method for factoring
this method starts out by multiplying the first and last coefficients
jim_thompson5910
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then you find two numbers that multiply to that result and add to the middle number
lyzzy7
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so the formula is ax^2 + bx + c right?
jim_thompson5910
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that's a basic form of a quadratic, yes
jim_thompson5910
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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
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so that's why they call it the ac method
lyzzy7
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I cant find the two numbers that multiply to -150 and add to -19..
lyzzy7
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im too dumb..
jim_thompson5910
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no you're not
jim_thompson5910
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stop that
jim_thompson5910
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there's a long list to check, but luckily there's a shortcut
jim_thompson5910
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have you heard of the quadratic formula before?
lyzzy7
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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
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ah i'm sry to hear that, at least she'll be gone and a (hopefully) better teacher will take over
lyzzy7
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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
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dang that's quite pricey
jim_thompson5910
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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)
lyzzy7
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i know right, and okay shoot
jim_thompson5910
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the quadratic formula is
\[\Large x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
does that formula look familiar?
lyzzy7
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noo.......
jim_thompson5910
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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)?
lyzzy7
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i guess so.
jim_thompson5910
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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.
lyzzy7
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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...
lyzzy7
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like i cant even read it in my head let alone get it. i have a headache.
lyzzy7
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fml. i hate math.
jim_thompson5910
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alright, sorry jumped too far ahead
lyzzy7
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its not your fault. it just is ridiculous that something that complicated even exists. i hate this world.
jim_thompson5910
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well much more complicated things exist (that even I don't understand), but that's what keeps things interesting
jim_thompson5910
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let's ignore the quadratic formula until you get more practice with it
lyzzy7
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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
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why not sleep then come back?
jim_thompson5910
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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)
lyzzy7
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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
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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
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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
lyzzy7
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Got it. and the length is 5x+3 right?
jim_thompson5910
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so
length = 5x + 3 (given)
width = 2x - 5 (just found this)
jim_thompson5910
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correct
lyzzy7
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ughhh now i have to solve two more, then ima sleep. thank you so much for the help!
jim_thompson5910
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sure thing
jim_thompson5910
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go ahead and post and hopefully it won't take too long
jim_thompson5910
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then you can finally sleep (til you have to get up in the morning...ugh lol)
lyzzy7
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you sure? i mean i could try to do them on my own if you want.
jim_thompson5910
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sure whatever works best for you
jim_thompson5910
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how about you work them and I check what you get
jim_thompson5910
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or I can show you how if you're stuck
lyzzy7
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alright heres this one
What is the factored form of 2x2 + x – 3?
then i'll do the other.
jim_thompson5910
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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
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hopefully I didn't cut you off there?
lyzzy7
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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
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lol magic
jim_thompson5910
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ah i should have let you say the two numbers, but hopefully you got 3 and -2 ?
lyzzy7
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yeah i did lol
jim_thompson5910
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ok perfect, you'll be a pro in no time
jim_thompson5910
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you just need to practice more with factoring by grouping (and the quadratic formula when you get more time)
lyzzy7
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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
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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
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nice work
lyzzy7
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OMG YAY:))) Thank you sooo much! like i really didnt get half my stuff till you helped lol
jim_thompson5910
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glad to be of help, now get some sleep lol
lyzzy7
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will do, thank you :)
jim_thompson5910
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I know, I probably sound like your mom, but idc lol
lyzzy7
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your good lol! how old are u by the way?
lyzzy7
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youre*
lyzzy7
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lol nah its cool. and its amazing that you put your brain to help people. what do you do as a job?
jim_thompson5910
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but thanks lol
lyzzy7
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i hope you go far! honestly
jim_thompson5910
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thanks, i hope so too
jim_thompson5910
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definitely getting a lot of practice here lol
lyzzy7
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lol, there ya go!
jim_thompson5910
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and building networks as well (well sorta i guess)
jim_thompson5910
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so that's another plus