some of the other things that we will learn will have formulas and things that you can memorize to potentially solve the equations faster if you happen to remember the formula but even if you've forgotten everything you can always use this method to solve any quadratic equation it works sort of like this let's try that with this ridiculous quadratic here i mean it has a number 899 in it but if i wanted to try to factorize this well it would have a form that has you know x inside it and x inside it as well and i'd like to factorize it so that the product equals zero i'm actually going to pre-write the factors with minus signs and the reason i want to write minus signs is if you have minus signs then it's particularly easy to read off the solutions because of the zero product property the zero product property told you that you'd have this product being zero precisely when at least one of these two factors was zero and x minus something is zero exactly when x is the something so if we were so lucky as to be able to get this factorization then those two would be the solutions okay so now my game becomes can i find two numbers to write here if you think about how factoring works and how foil works i would need to have that these two numbers multiply together to be the 899. i've already put the minus signs in front and the minus times the minus is the plus so i'm looking for two numbers that multiply to 899. have we memorized our multiplication table long enough to know where 899 appears actually no it's too big that's why this is hard to factor but remember the other piece you need is they need to not only multiply to 8.99 by the foil if you try to expand this out you will need that these two things add up to the 60. so i'll be able to find the factoring if i can find a way to find two numbers that multiply to 899 and add to positive 60. and we found out that the multiply to 899 is very difficult to figure out how to do and that's when we do something different you see instead of unmultiplying the 8.99 let's un-add the 60. can you think of two numbers that add to 60 oh that's easy here are two 30 plus 30. that's 60. other numbers that add to 60 well here's another two 20 plus 40. do you see a pattern here when i'm hunting for numbers that add to 60 there's a way to think of them where 30 and 30 that's like half of 60 plus half of 60 is the whole 60. and if i move apart from the 30 and the 30 by equal amounts in opposite directions then the sum is going to be still 60. so there's a hope a glimmer of hope i want to know does there exist an amount that i can move by away from the 30 that works i'll write it like this does there exist some unknown number u u stands for unknown so that if i move down by u that's 30 minus u and i also move up by u that's 30 plus u that somehow this multiplies to 899. this would make me really happy if i could find because then i would have these two numbers multiplying together to get me my 899 and they would actually add together to give me my 60. so all i have to do to solve this quadratic equation is solve another even more messy quadratic equation oh no except that a miracle occurs if i'm trying to solve this what happens as you try to expand this left side let's do the foil thing 30 times 30 that's 900 okay 30 and plus u gives me plus 30 u minus u and 30 gives me minus 30 u and then the minus u and the plus u gives you minus u squared this is the equation here now and i'm just going to write some arrow saying if i was able to find a u that would work here then it would work there too you see i call this a miracle because there's a 30 u and a 30 u with opposite signs so actually this part here is zero this particular form of something minus another thing times the first thing plus the other thing that always expands into what's called the difference of squares and that doesn't have this nasty middle term now let's think could you find some way so that 900 minus something is 8.99 well sure the way to do that is if the something is one if i was able to find a u so that u squared is 1 then 900 minus u squared that's 8.99 and if this makes you uncomfortable another way that you can see this is if you subtracted 899 from both sides that's one here and you added u squared to both sides that would get rid of it here and put a u squared there it would give you 1 equals u squared which is the same oh can you think of any numbers that's square to 1 i can one actually minus 1 also we'll think about that in a second but i'm writing the arrow going from down to up meaning that if i have u equals 1 oh yeah u squared equals 1. and this is true and that is true oh wait then that's a factoring oh so u equals 1 that means i want 30 minus 1 and 30 plus 1. that's 29 and 31 really does it work let's see and we'll talk about the minus one in a second do 29 and 31 add to 60 they sure do what about multiplication that's hard i haven't done multiplication in a while never ask a math professor to do multiplication 31 times 29 what is this 9 times 1 is 9 9 times 3 is 27 2 6 and i think that's how you multiply oh my gosh 899 it actually works so that means that this is a factorization that means that those two are solutions what about that minus 1 if i had used minus 1 because i just decided that i like minus one as a number that squares to one what's 30 minus minus one 31 what's 30 plus minus 1 29. so it's the same two just in a different order math is consistent we're still getting the two solutions bam we've just managed to factorize this let's do one more which is even more nasty this now it doesn't look like fun for factoring because there's this two sticking in front well the good thing is in algebra the way i think of algebra there's a basic rule if you got an equation there's an equal sign you're allowed to beat up the equation as much as you want as long as the left and the right sides get beaten up equally badly and so what i want to do is i just want to divide both sides by 2. and so as i divide both sides by 2 this gives me another equation x squared minus 8x plus 13 equals to 0. for those of you who are curious about the logic here these two are actually equivalent equations because i can go from the top one to the bottom one by dividing both sides by two and i can go from the bottom one to the top one by multiplying both sides by two let's go and play the factoring game now well i would like to know i like minus signs i would like to know can i find some numbers let's leave some space here some numbers i promise you these will be numbers some numbers that i can stick here so that if i expand it out i get exactly the left side well now my task is a little bit more complicated i need to find two numbers here whose sum is equal to eight remember i have minus signs already and whose product is equal to 13. good luck finding two numbers whose product is equal to 13 because 13 is prime and so like 1 and 13 uh-oh that doesn't add to 8. but let's do this trick again let's un-add the 8. so now i'm just curious does there exist a u where i can move equal distance from half of 8. does there exist a u such that remember from half of 8 so i need 4 minus u times 4 plus u to be 13. if i'm so lucky as to find this u then suddenly i have two numbers who add to eight two numbers which multiply the 13 because that's what this says and then suddenly i get to use the zero product property and the factoring and those two will be the solutions but let's do it again now you see that's this difference of squares pattern again this is just 16 minus u squared and if you remember that's because this would have gotten a 4u and a minus 4u which cancel oh how do you do this 16 minus something equals 13 well then that happens exactly when the something is 3. and again if you prefer to do this a bit more slowly you could think of that by adding u squared to both sides so there's a u squared here this is gone and subtracting 13 from both sides so this becomes 3 and that's just this guy backwards do you know any numbers which square to 3 well that's what the square root of 3 is square root 3 works and that would give me something that squares to 3. minus square root 3 works also but this is enough for my purposes because look 4 minus root 3 and 4 plus root 3. so actually my factoring is this horrible looking thing 4 minus root 3 and 4 plus root 3. let's erase these boxes so that i can see it more clearly this is actually a factorization of that how do you know well i mean if we check with the foil if i add these two things together i will get 8. and if i multiply these together what's 4 minus root 3 times 4 plus root 3. well that's why we did this right 4 minus something 4 plus something is 16 minus the something squared and so that's actually 16 minus 3 which is 13. it works so what that means is that the solutions to this particular quadratic equation are exactly 4 minus root 3 and 4 plus root 3. we mathematicians are lazy we don't want to write that twice so we write that as 4 plus or minus root 3. whenever you see a plus or minus sign that just means that one option is with the plus one option is with the minus and that's how you solve this crazy quadratic would anyone have been able to just guess this by doing the factoring probably not that easily because how would you ever think of guessing these to multiply together to make the 13. well what you've just seen here is actually a technique that you can use to factor any quadratic if the first coefficient here is not 1 like this you can just divide it out the numbers are nicer when after dividing out the second coefficient happens to be an even number but the reason i wanted to share this method is because it actually requires almost no memorization in fact all you have to do is just think about oh yes there's a factoring method there was this game to find out given the sum and the product can i find the two numbers actually i came up with this observation a few years ago as i was playing around with uh with quadratic equations and when i realized you could use this to factor any quadratic i almost fell out of my chair because i was like how how in the world was this not widely known then i went digging and i found out actually this method of how to break apart summon product that was known like two or three thousand years ago by the babylonians but somehow it didn't make it into our algebra books why here's some philosophy if you want to know why the babylonians might have used this funny method of unadding instead of unmultiplying let's go back to this 899. for us we don't know where 899 is in the times table because we don't have that timestable that gets that big we use base 10 and that means that our digits are 0 1 2 3 4 5 6 7 8 and 9. and so our times table that we memorize is only from like 1 to 10 to 1 to 10. babylonians use something called base 60. so actually their times table would have memorizing like 60 by 60 some massive thing i mean fourth grade would have taken like five years you know so so the point is that babylonians actually didn't have a times table memorized therefore actually if you don't have the times table memorized the more natural way to approach this problem is not to unproduct the product but to unsum the sum so anyway some of the other things that we will learn uh will have formulas and things that you can memorize to potentially solve the equations faster if you happen to remember the formula but even if you've forgotten everything you can always use this method to solve any quadratic equation there is no reason why everyone should have access to the very best education welcome to calculus one to introduction to astronomy the introduction of philosophy to statistics microeconomics it's a culture let's get started [Music] you