Solving a quadratic by factoring

welcome to solving a quadratic by factoring let's start doing some problems so let's say I had a function f of X is equal to x squared plus 6x plus 8 now if I were to graph f of X the graph is going to look something like this I don't know exactly what it's going to look like but it's going to be a parabola and it's going to intersect the x axis at a couple of points here and here and what we're going to try to do is determine what those two points are so first of all when a function intersects the x axis that means f of X is equal to 0 because this is the f of X axis similar to the y axis so here f of X is 0 so in order to solve this equation we set f of X to 0 and we get x squared plus 6x plus 8 is equal to 0 now this might look like you could solve it pretty easily but that x squared term kind of messes things up and you could try it out on your for yourself so what we're going to do is factor this and we're going to say that x squared plus 6x plus 8 that this can be written as X plus something times X plus something and will still equal that so that's equal 0 now in this presentation I'm going to show you the the systematic or you could say the mechanical way of doing this so I'm gonna give you another presentation on on why this works and and you might want to just multiply out the answers we get and multi multiply out the expressions and see why it works and the method we're going to use is we look at the coefficient on this X term and 6 and we say what two numbers will add up to 6 and when those same two numbers are multiplied you're going to get 8 well let's just think about the factors of 8 the factors of 8 are 1 2 4 & 8 well 1 times 8 is 8 but 1 plus 8 is 9 so that doesn't work 2 times 4 is 8 and 2 plus 4 is 6 so that works so we could just say X plus 2 and X plus 4 is equal to 0 now if two expressions or two numbers times each other equals zero that means that one of those two numbers or both of them must equal zero so we have two now we could say that X plus 2 equals 0 or or and X plus 4 is equal to 0 well this is just a very simple equation we subtract 2 from both sides and we get x equals negative 2 and here we get x equals minus 4 and if we substitute either of these into the original equation we'll see that it works minus 2 so let's let's just try it with minus 2 and now leave it - 4 up to you so minus 2 squared plus 6 times minus 2 plus 8 minus 2 squared that's a squared is 4 minus 12 6 times minus 2 plus 8 and sure enough that equals 0 and if you did the same thing with negative 4 you'd also see that that works and you might be saying wow this is interesting this is an equation that has two solutions well if you think about it makes sense because the graph of f of X is intersecting the x axis in two different places let's do another problem let's say I had f of X is equal to 2x squared plus 20x plus 50 so if we want to figure out where two intersects the x-axis we just set f of X equal to zero and I'll just swap the left and right left and right sides of the equation and I get 2x squared plus 20x plus 50 equals zero now what's a little different this time from last time is here the coefficient of x squared is actually two instead of a one and I like it to be a 1 so let's divide the whole equation both the left and right sides by 2 and I get x squared plus 10x plus 25 equals zero so all I did is I multiplied one half times this is the same thing as dividing by two and did times one half and of course zero times one half is 0 now we're ready to do what we did before and you might want to pause it and try it yourself we're going to say X plus something times X plus something is equal to zero and those two nuts some things they should add up to 10 and when you multiply them they should be 25 let's think about the factors of 25 you have 1 5 and 25 well 1 times 25 is 25 but 1 plus 25 is 26 not 10 5 times 5 is 25 and 5 plus 5 is 10 so 5 actually works so actually turns out that both of these numbers are 5 and so you get X plus 5 equals 0 or X plus 5 equals 0 so you just have to really write at once so you get x equals negative 5 so how do we think about this graphically I just told you that these equations can intersect the x-axis in two places but this one only has one solution well this solution would look like this if this is x equals negative five we'd have a parabola that just touches right there and then it comes back up so instead of intersecting two places it only intersects right there at x equals negative five and now as an exercise just to prove to you that I'm not I'm not teaching you incorrectly let's let's multiply X plus 5 times X plus 5 just to show you that it equals what it should equal so we just say that this is the same thing as x times X plus 5 plus 5 times X plus 5 X 10 so you get x squared plus 5 X plus 5 X plus 25 and that's x squared plus 10x plus 25 so it equals what we said it should equal and I'm going to once again do another module where I explain this a little bit more let's do one more problem and this one I'm just going to cut to the chase let's just solve x squared minus X minus 30 is equal to zero once again two numbers when we add them they equal what's the coefficient here it's negative one so we can even say we could say those two numbers or a plus B equals minus one and a times B will equal minus 30 well let's just think about what all the factors are of 30 there's one two three five six ten fifteen and thirty well something interesting is happening this time though since a times B is negative thirty one of these numbers have to be negative they both can't be negative because if if they're both negative then this would be a positive 30 so eight times B is negative thirty so it's actually we're going to say two of these factors the difference between them should be negative one well if we look at all of these all these numbers obviously when you pair them up they multiply out to 30 but the only ones that have a difference of one is five and six and since it's a negative one it's going to be and I know I'm going very fast to this and I'll do more example problems so it'll be X minus 6 times X plus 5 is equal to 0 so how did I think about that negative 6 times 5 is negative 30 negative 6 plus 5 is negative 1 so it works out and the more and more you do these practices I know it seems a little confusing right now it'll make a lot more sense so you get x equals 6 or x equals negative 5 I think at this point you're ready to try some some solving quadratics by factoring and I'll do a couple more modules as soon as you get some more practice problems have fun