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0:00
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
0:30
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
0:53
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
1:26
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
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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
2:25
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
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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
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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
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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
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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
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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
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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
6:25
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
6:50
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
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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
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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
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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
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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
9:17
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
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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
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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
10:55
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
11:26
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
11:57
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
12:34
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
13:00
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
13:29
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
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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
14:24
-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
15:09
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
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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
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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
16:36
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
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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
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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
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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
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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
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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
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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
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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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