Question

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

Answer 1

@jim_thompson5910

Answer 2

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

Answer 3

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

Answer 4

yes please @ParthKohli

Answer 5

8*? has to equal 16

so basically

8p = 16

p = ???

Answer 6

where p is that answer

Answer 7

2

Answer 8

yep, you nailed it

Answer 9

yayyy :)

Answer 10

so

v^2 + 10v + 16 = (v + 8)(v + ?)

turns into

v^2 + 10v + 16 = (v + 8)(v + 2)

Answer 11

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

Answer 12

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.

Answer 13

find two numbers that

a) multiply to 8 (last term)

AND

b) add to 6 (middle coefficient)

Answer 14

im confused @ParthKohli

Answer 15

2 and 4 . @jim_thompson5910

Answer 16

It's simple.

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

Answer 17

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

Answer 18

ohhh! @ParthKohli thanks :)

Answer 19

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

Answer 20

I know that was sarcastic. -_-

Answer 21

hold on im thinking @jim_thompson5910

Answer 22

5 and 3?? @jim_thompson5910

Answer 23

you found the numbers already by saying 2 and 4

2+4 = 6 (middle coefficient)

2*4 = 8 (last term)

Answer 24

ohh.

Answer 25

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)

Answer 26

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

Answer 27

go ahead and expand out (x+2)(x+4)

you'll find that you'll get x^2 + 6x + 8 back again

Answer 28

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

Answer 29

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

Answer 30

correct, the order of the factors doesn't matter

so (x+2)(x+4) is the same as (x+2)(x+4)

Answer 31

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

Answer 32

sure go for it

Answer 33

What is the factored form of x^2 – 7x + 12?

and dont worry i knew what you meant lol

Answer 34

same idea, just different numbers

Answer 35

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

Answer 36

-3 and -4

Answer 37

perfect

Answer 38

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

Answer 39

(x – 4)(x – 3) ?

Answer 40

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

Answer 41

you got it

Answer 42

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

Answer 43

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?

Answer 44

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

Answer 45

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

Answer 46

what do you get when you do?

Answer 47

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

Answer 48

like for the question

Answer 49

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

Answer 50

-6 and -4

Answer 51

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

Answer 52

so try again

Answer 53

oops lol hold on

Answer 54

-6 and 4?
-6*4= 24
-6+4 = -2

i think lol

Answer 55

good

Answer 56

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

Answer 57

notice how g+4 is a factor

Answer 58

the other factor is g-6

Answer 59

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?

Answer 60

g-6!

Answer 61

correct since Area = Length times Width

Answer 62

:D

Answer 63

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

Answer 64

I actually get that!

Answer 65

it only works for rectangles though

Answer 66

that's great

Answer 67

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

Answer 68

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

Answer 69

but we can still use some of the trick

Answer 70

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

Answer 71

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

Answer 72

2 and 10

Answer 73

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

Answer 74

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)

Answer 75

hopefully factoring by grouping is a familiar term

Answer 76

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

Answer 77

correct

Answer 78

hopefully you see how I got all that

Answer 79

okay, thanks :D and lol yeah mostly.

Answer 80

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?

Answer 81

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

Answer 82

10*(-15) = -150

Answer 83

find two numbers that multiply to -150

and

they add to -19

Answer 84

why do you multiply 10 and -15?

Answer 85

because I'm using the AC method for factoring

this method starts out by multiplying the first and last coefficients

Answer 86

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

Answer 87

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

Answer 88

that's a basic form of a quadratic, yes

Answer 89

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

Answer 90

so that's why they call it the ac method

Answer 91

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

Answer 92

im too dumb..

Answer 93

no you're not

Answer 94

stop that

Answer 95

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

Answer 96

have you heard of the quadratic formula before?

Answer 97

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.