Factoring quadratics with common factor (old)

factor eight K squared minus 24 K minus 144 now the first thing we can do here just eyeballing each of these terms if we want to simplify it a good bit is all of these terms are divisible by 8 clearly 8k squared is divisible by 8 24 is divisible by 8 and 144 it might not be as obvious is divisible by 8 but it looks like it is 8 goes into 144 8 goes into 14 one time 1 times 8 is 8 subtract you get a 6 14 minus 8 is 6 bring down the 4 8 goes into 64 eight times so it goes into 144 18 times so let's just factor out an 8 of this and then that'll simplify our expression lecture give us a leading 1 coefficient so this will become this will become 8 times K squared minus 24 divided by 8 is 3 K - -18 - 18 now we have to factor this business in here and remember if anything has the form x squared plus BX plus C where you have a leading 1 coefficient this is implicitly a 1 we have that here in this expression in parentheses then we literally just need a and we can we can do this multiple ways but we need to find two numbers whose sum is equal to the coefficient on X so two numbers whose sum is equal to negative 3 and whose product whose product is equal to the constant term and whose product is equal to negative 18 so let's just think about the factors of negative 18 here let's see if we can do something interesting so it could be 1 and if it since it's negative one of the numbers has to be positive 1 has to be negative 1 and 18 if is if it was positive and then one of these could be positive and then one of these could be negative but no matter what we're either going to have we're gonna if you if this is negative and this is positive then this will be they add up to 17 if you switch them then they add up to negative 17 so those won't work so either we could write it this way positive or negative 1 and the negative or positive to show that they have to be different signs so those don't work then you have positive or negative three and the negative or positive six just to know that there are different signs so if you have positive three and negative six they add up to negative three which is what we need them to add up to and they're clearly positive three and negative six their product is negative eighteen so it works so we're going to go with positive three and negative six as our two numbers now for this example just for the sake of this example we'll do this by grouping will do this by grouping so what we can do is we can separate this middle term right here as the sum of 3k and negative six K so I can write the negative 3k as plus 3k minus 6 K and then let me write the rest of it so we have K squared up here plus 3k minus 6 K which is the same thing as this over here and then we have minus 18 and then all of that is being multiplied by 8 now we're ready to group this thing we can group these first two terms they're both divisible by K and then we can group we put a positive sign let's group these second two terms so then we have 8 times all the right brackets here instead of writing double parentheses brackets are really just parentheses that look a little bit more serious okay now let's factor out let's factor out a K from this term right over there just in a different color let's factor out the K here so this is K times k plus 3 and then we have + and then over here it looks like we could factor out a negative 6 so let's factor out and do this a different color let's factor out a negative 6 over here so plus negative 6 times k plus 3 times k plus 3 so now it looks like we can factor out a k plus 3 there's a capable of 3 times the K and then we have a k plus 3 times the negative 6 so let's factor let's factor that out so we have this 8 out front that's not changing so let me write that in the brackets we're factoring out a k plus 3 so that we have the k plus 3 that we factored out and then inside of that we just have left this K instead of writing plus negative six I could just write k minus 6 K minus 6 K minus 6 we factor out the K plus 3 and we're done and then we can rewrite this is the way we were already here it's 8 times the product of K plus 3 times K minus 6 but we know from the properties of multiplication this is the exact same thing as 8 times k plus 3 times k minus 6 and we are done