I came across this amazing new factoring method  that you've probably never seen before if you  

watch to the end of this video you'll be able to  factor absolutely any type of quadratic expression  

without having to guess and check any values  for your product and sum let me show you how  

we can Factor the standard form quadratic x^2 +  32x + 192 without having to do any guessing and  

checking now when your teacher first taught you  how to factor this type of quadratic trinomial  

that has an a value of one a b value of 32 and a  c value of 192 your teacher probably taught you  

that you can split this quadratic into two factors  where in each of those two factors they start with  

an x and then you would have to do some sort of  guessing and checking until you could figure out  

what numbers you add to each of those X's to make  this product equal that standard form quadratic  

now you wouldn't just guess and check any random  numbers you probably learned that these numbers  

let's call them m and n those values of M and N  always have a sum of the B value so in this case  

32 and they also have a product of the C value  192 and at this point you would probably start  

guessing and checking until you can find the pair  of numbers M and N that satisfy this product and  

sum and how students in my class usually do this  is they start focusing in on the product and they  

think of what numbers multipli to 192 well to 2 *  96 is 192 but 2 + 96 is not 32 so 2 and 96 don't  

work and then students keep guessing and checking  numbers that have a product of 192 until they find  

the ones that also have a sum of 32 so maybe they  would scroll through a few more options until they  

actually find them so they could try 4 * 48 that's  also 192 but those don't add to 32 and 12 and 16  

multiply to 192 but they don't add to 32 and you  can see how this guessing and checking can get  

timec consuming and it really relies on students  being good at knowing the factors of 192 so there  

has to be a better way where we don't have to  go through this guessing and checking let me  

show you that so let's array solve this and let's  find those product and sum numbers without having  

to guess and check anything and how we're going  to do that we're going to focus instead on the  

sum being 32 well half of 32 is 16 so for that  reason I know two 16s added together would be  

32 and in fact any numbers that have a sum of 32  the average of those numbers is going to have to  

be 16 so that when they're added together we get  32 so for example 15 + 17 is also 32 notice 15 is  

one less than 16 and 17 is 1 more than 16 and if  I continue this pattern 14 + 18 would also be 32  

14 is 2 less than 16 and 18 is two more than 16 so  the two of those numbers have an average value of  

16 which is why when we add them together we get  double 16 which is 32 and let me write one more  

sum so just you get this principle 13 which  is 3 less than 16 if I add that to a number  

that's three more than 16 which is 19 I'll get  32 so to find a sum of 32 the two numbers that  

I pick one of them is going to be less than 16  and one of them is going to be greater than 16  

but the distance that they are from 16 is going to  be equal to each other so I could write a general  

rule for any two numbers that have a sum of 32  I could say that 16 minus some constant plus 16  

plus that same constant would have to equal 32 as  long as I move the same amount below 16 and above  

16 I know that the sum of those two numbers is  going to be 32 because the average of these two  

numbers is going to be 16 and now that we have  this general rule where we could say 16 minus  

C is our M value and 16 plus C is our n value I  know that these two values M and N have to have  

a product of 192 so I can set up that product  16 minus C * 16 + C has to be equal to 192 and  

now I just have to solve for that constant C this  product is a difference of squares so following  

the difference of squares formula I would have  60 16 ^ 2 - c^ 2 is equal to 192 well 16 2 is  

256 and if I isolate c^ 2 256 - 192 is 64 and then  to solve for C I could do plus or minus the square  

root of 64 so there are technically two answers to  this the square Ro T of 64 is 8 so our two answers  

are plus or minus 8 but we actually don't need  both of these answers we only need one of them  

no matter whether we subed in positive 8 or  negative 8 in for C into these two factors  

we'll end up with the same two values for M and  N just in a different order and the order doesn't  

matter so why don't we just take the positive  square OT of 64 to make our answer for C 8 and  

if C is 8 if I sub it into this expression for the  sum of the two numbers that equal 32 I would have  

16 - 8 + 16 + 8 = 32 which means we have 8 + 24 =  32 which it does 8 + 24 is 32 so this m value of8  

and this n value of 24 has the right sum it adds  up to 32 but do they also have the right product  

well if I check 8 * 24 is 192 so we were able  to get those right numbers 8 and 24 that have  

the write product and sum without having to do  any guessing and checking so to write the final  

factored form I just take this m and n value and  replace the m and n in the factored form that I  

have written here so my final answer is just x + 8  * x + 24 now having seen this method for the first  

time hopefully you were able to follow it and gain  a bit of an appreciation for how intuitive and  

amazing this method is for being able to get this  quadratic into factored form without having to do  

any guessing and checking for product and sum  numbers but you might be a little bit skeptical  

this quadratic was set up fairly easily to be able  to work well with this method but I guarantee you  

this method actually works to factor any quadratic  expression even quadratics where the a value is  

not one and also quadratics that are going to  have to have irrational or imaginary numbers in  

their factors so keep watching till the end and  I'll show you all of those different types of  

quadratics and we'll see how this method works for  all of them you'll be able to factor absolutely  

any quadratic without any guessing and checking  by the end of this video so let's try another  

example here's another fairly straightforward  one just to get you comfortable with the method  

we're going to factor the quadratic x^2 - 6x - 91  once again we have an a value of 1 B value of -6  

C value of 91 I know this is going to be able to  split into two factors each of those factors will  

start with an x and to each of those X's I'm  going to add two values we call them M and N  

but we're going to have to do some work to find M  and N because the quadratic has an a value of one  

I know that M and N will have a sum of the B value  so they'll add to -6 and they'll have a product  

of the C value 91 and now get out of the habit  of guessing and checking numbers that multiply  

to 91 and then trying to find ones that also have  that sum we're going to instead start by thinking  

about this sum in order for numbers to have a sum  of -6 I know those two values M and N are going  

to have to have an average value that is half of  -6 so the you're going to have to have an average  

value of -3 in order for two numbers to have an  average of -3 I could add some constant to neg3  

to get my first number as long as my second  number is that same distance C but less than  

-3 so I would have to subtract C from my second  number and because each of these values are an  

equal distance greater than -3 and less than -3 I  know that when I add them together they're going  

to equal -6 let me just shrink that a bit and  now that I can associate --3 + C with M and -3  

minus C with n I know I want those M and N values  these two values here to also have a product of 91  

so I can set up that product and then this is a  difference of squares it would equal 9 - c^2 if  

I expanded it isolating c^2 on the left I'll have  9 + 91 which is 100 equal to c^2 and then I just  

need one solution to this equation so I'll just  get the positive otk of 100 which is 10 now that  

I know that c is 10 I can sub it back into these  two equations to make sure it satisfies my product  

and sum correctly into this sum equation I would  have -3 + 10+ -3 - 10 is equal to -6 which means  

my two numbers are 7 and -13 when I add those I  do get -6 and If I multiply them if I do 7 * -3  

13 I do get 91 so I have found my correct product  and sum numbers 7 and -3 without having to do any  

guessing and checking I just take these two values  and replace M and N with them and then I have my  

correct factored form so the factored form is  x + 7 * x + -13 which we would just write as  

x -3 now let's try one that's a little bit more  difficult you'll notice in the first two examples  

the sum that we wanted when I took half of that  number I was able to get an integer value instead  

of a fraction let's see what happens if we get a  fraction when we find half of the number we want  

a sum of let's say we want to factor the quadratic  expression x^2 + 9x + 20 this quadratic still has  

an a value of 1 as a b of 9 and a c of 20 because  of the a value of one I know it's going to factor  

to two factors that both start with X and I'll  have to find the two numbers that I add to those  

X's we'll call them M and N so that this factored  form equals the standard form now I know M and N  

have to have a sum of the B value 9 and they  have to have a product of the C value 20 now  

the numbers that satisfy this product and sum are  actually pretty obvious that it's four and five  

that work let me show you how the algorithm I've  taught you will work for this as well we start  

by focusing in on the sum and I know that if two  numbers have a sum of N the average of those two  

numbers has to be half of n which I could write as  9/2 so my M value could start at 9/2 and I could  

add any constant I want to that as long as for my  n value I subtract that same constant from 9/2 so  

that these two numbers have an average value of  9 over2 so that when I add them I get nine shrink  

that a bit and now I know since these are my M and  my n I know they have to have a product of 20 and  

then expanding this product I would get 9 over 2^  2 minus c^ 2 9 over 2^ 2 is 81 over 4 if I isolate  

c^2 I've got 81 over4 minus 20 I could rewrite 20  as 80 over 4 and 81 over4 - 80 over 4 is 1 over4  

and then C would be equal to the square < TK of  1 over 4 the S > of 1 is 1 and theun of 4 is 2  

so my C value is a half take take this C value and  sub it into my sum and my product equation to make  

sure I have found the correct values of M and N  starting with the sum equation if I sub in my C  

value of a half I have 9 /2 + a half + 9 / 2 - a  half 9 /2 + 1 over 2 is 10 / 2 which is 5 and 9 /  

2 - 1 over 2 is 8 over2 which is 4 so 5 + 4 is 9  and if I check the product of course 5 * 4 is 20  

so I have the correct sum and the correct product  so I can just take those values and replace M and  

N with it which means my factored form is x + 5 *  x + 4 now that we've done three examples where the  

a value is one let's try an example where the a  value is not one so you can see how it works let's  

Factor the quadratic 2x^2 - 7 x - 4 notice the a  value of this quad quadratic is not one but the  

factoring method I've been showing you only works  for quadratics that have an a value of one so what  

we have to do is start by Common factoring out  this two from all three terms of this quadratic so  

I'll remove a common factor of two and then I'll  divide all three terms by 2 giving me x^2 - 7/  

2x - 2 this quadratic has an a value of 1 a b  value of -7 over2 and a c value of -2 so because  

the a value is one I know I'm going to be able to  split that quadratic into two factors where each  

of the factors starts with an X I'm just going to  have to figure out what are the m and the N values  

that I add to those x's and I know that the sum  of M and N has to be equal to the B value of -7  

over2 and the product of M and N has to equal the  C value of -2 and I'll start by focusing on the  

sum I know that the sum of two numbers could be  -7 /2 as long as those two numbers both start at  

half of -7 /2 which is -7 over4 and to the first  one I can add any constant I want as long as to  

the second number in my sum I subtract that same  constant that will make the sum still be -7 over  

2 and now that I know that these are my M and my n  values I can set my product of those two equal to  

-2 and then in this equation if I expand the left  side of it I would get 49 over 16 - c^2 = -2 and  

then isolating c^2 I'd have 81 over 16 = c^2 which  means C is the square root of that which is 9 over  

4 now that we have the C value we can solve for M  and N by subbing into these two expressions after  

subbing in I find out that my two numbers  are a half and -4 a half + -4 is -3 and A2  

and checking my product a half * -4 is -2 so we  have the correct M and N values so I can write my  

correct factored form my final answer is just 2 *  x + a half * x + -4 which we would just write as  

x - 4 and for your factors to be fully simplified  you shouldn't have any fractions in your factors  

so what we can do is distribute this two to that  first factor to take care of that problem which  

give us 2x + 2 * a half is 1 so 2x + 1 * x  - 4 there's our final factored form of this  

quadratic and there is no guessing or checking  necessary let's now have a look at this quadratic  

x^2 + 8x + 4 this quadratic has an a value of 1 B  value of 8 C value of 4 if you're going to try and  

Factor this we know we could split this into two  factors where because the a value is one each of  

these factors will just start with X but then we  have to figure out what M and N values need to be  

added to those X's to get it into factored form  well remember the m and the N would have to add  

to the B value and they would have to multiply to  the C value and this is another great example of  

why this method is so good because if you were  to try and guess and check numbers that have a  

product of four and a sum of eight there are no  integers that work so your teacher would probably  

tell you just to write that this quadratic is  not factorable but we can actually get it into  

factored form using this method mind you within  the factors we're going to have some irrational  

numbers but that's fine we will be able to get  it into factored form we start by looking at the  

sum of eight and I know two numbers will have a  sum of eight if both of those numbers start at  

half of eight which is four and then as long as  I add and subtract the same amount from each of  

those fours the sum will still be eight make this  a bit smaller and then we'll focus on our product  

being four since my product has to be four I  know that 4 + C * 4 minus C is going to have  

to be four and then I can solve this equation if  I expand this product it's a difference of squares  

it would be 4^ 2 which is 16 - c^ 2 16 - 4 is 12  so 12 is equal to c^2 and C would equal we'll just  

get the positive < TK of 12 which could be split  into < TK4 * < tk3 but the < TK of 4 is 2 so I  

could simplify this to 2 < tk3 now that I have my  C value I can write my sum and product with those  

C values subbed in to check and make sure it works  since we have found the numbers that satisfy both  

the sum and the product I can take those numbers  for m and n and replace the m and the n in my  

factored form with them and there's my factored  form of this quadratic and let me give you one  

more final example where we have a quadratic that  we can only factor using a maginary numbers let's  

look at this quadratic x^2 + 4x + 10 the a a value  is 1 B is 4 C is 10 because the a value is 1 I'll  

be able to split this into two factors where each  of those two factors start with an x but I'm going  

to have to figure out the m and the N values that  I add to those X's to get it into factored form  

I know that the m and the N values have to add to  the B value four and they have to multiply to the  

C value 10 if you try to guess and check numbers  that satisfy this product and sum you'd probably  

give up pretty quickly and just say that this  quadratics not factorable but we in fact can get  

it into factored form let me show you how we'll  focus on the sum the sum would be four if I add  

two numbers that both start at half of four which  is two but to each of those two numbers I have  

to add and subtract the same amount and that will  ensure that the sum stays at four shrink that and  

now that I have my M and my n values I can make  those have a product of 10 I'll expand the left  

side of this equation to give me four - c^ 2 which  means that -6 = C ^2 and then we'll only consider  

the positive square root which means C = theare <  TK of -6 which we can actually write as a product  

of theare < TK of -1 and theare < TK of 6 and the  < TK of1 we have a symbol for that we could just  

use the symbol I to represent theun of1 now that I  have my C value I can determine what my M and my n  

values are that have the correct sum and product  so I have found the m and the N values that have  

the correct product and sum so I'll just sub them  in for m and n and then I have my factored form  

equation all right hopefully you've enjoyed  this video and you now have a new method for  

factoring any quadratic without having to spend  any time guessing and checking values for your  

sum and your product leave me a comment letting  me know if you've ever seen this method before  

and also make sure to show your teachers this  method I'm sure they'll appreciate it Jensen a

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Factoring Quadratics WITHOUT Guessing Product & Sum