Factoring quadratics: common factor + grouping

we're asked to factor 35 K squared plus 100 K minus 15 and because we have a non zero code or non-one coefficient out here the best thing to do is probably buy is probably to factor this by grouping but before we even do that let's see if there's a common factor across all of these terms and maybe we can get a 1 coefficient out there if we can't get a 1 coefficient well at least have a lower coefficient here if we look at all of these numbers they all look divisible by 5 in fact they are all their greatest common factor is 5 so let's at least factor out of 5 so this is equal to 5 times 35 K squared divided by 5 is 7 K squared 100 K divided by 5 is 20 K 20 K and then negative 15 divided by 5 is negative 3 so we were able to factor out a 5 but we still don't have a 1 coefficient here so we're still going to have to factor by grouping but at least the numbers here are smaller so it'll be easier to think about it in terms of finding numbers whose product is equal to 7 times negative 3 and whose sum is equal to 20 so let's think about that let's figure out two numbers that if I were to add them or even better if I were to take the product I get 7 times negative 3 7 times negative 3 is equal to 7 times negative 3 which is equal to negative 21 and if I were to take their sum if I add those two numbers it needs to be equal to 20 it needs to be equal to 20 now once again because their product is a negative number that means they have to be of different signs so when you add numbers of different signs you could view it as you're taking the difference of the positive version so the difference between the positive versions of the number has to be 20 so the number that immediately jumps out is we're probably going to be dealing with 20 and 21 and 1 will be the negative because we want to get to a positive 20 so let's think about it so if we think of 20 and negative 1 their product is negative 21 sorry if we take 20 21 21 and negative 1 their product is negative 21 21 times negative 1 is negative 21 and if you take their sum 21 plus negative 1 that is equal to 20 so these two numbers right there fit the bill now let's break up this 20k right here into a 21 K and a negative 1 K and a negative 1 K so let's do that so let's rewrite the whole thing we have 5 times 7 K squared and I'm going to break this 20 K I'm going to break that 20 K into a let me do it in this color right here I'm going to break that 20 K into a plus 21 K minus K right that's or you can say minus 1 K if you want I'm using those two factors to break it up and then we finally have the minus 3 right there now the whole point of doing that is so that we can now factor each of each of the two groups this could be our first group right here and so what what can we factor out of that group right there well both of these are divisible by 7 K so we can write this as 7 K times 7 K squared divided by 7 K you're just going to have a K left over and then plus 21 K divided by 7 K is just going to be a 3 so that factors into that and then we can look at this group right here they have a common factor well we can factor out a negative 1 if we like so this is equal to negative 1 times K divided by negative 1 is K negative 3 divided by negative 1 is positive 3 and of course we have this 5 sitting out there we have this 5 sitting out there the whole time now ignoring that 5 for a second you see that both of these inside terms both of these inside terms have K plus 3 as a factor right this have this K plus 3 as a factor so we can factor that out so let's ignore this 5 for a second this inside part right here the stuff that's inside the parentheses we can factor k plus 3 out and it becomes K plus 3 times k plus 3 times 7 K 7 K minus 1 minus 1 and if you don't if this seems a little bizarre to you just tribute the K plus 3 on to this k plus 3 times 7 K is that term K plus 3 times negative 1 is that term and of course the whole time you have that 5 sitting outside you have that 5 we only have to put parentheses there says five times k plus 3 times 7 K minus 1 and we factored it we're done