Learn about Solving a quadratic by factoring Video

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Learn about 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 what exactly it’s going to look like, but it’s going to be a parabola and it’s going to intersect the X axis and then a couple of points here and here, and what we’re going to try to do is determine what those 2 point are. So first of all when a function intersects the X axis that means F of X is equal to zero because this is the F of X axis somewhere to the Y axis, so here F of X is zero so in order to solve this equation we set F of X to zero and we get X squared plus 6X plus 8 is equal to zero. Now this might look like you could solve it pretty easily but that X squared term kinda messes things up and you could try it out on, for yourself. So 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 we’ll still equal that, so that’s equal zero. Now in this presentation I’m going to just show you the , the systematic or you could say the mechanical way of doing this, now I’m going to give you another presentation on why this works and you might wanna just multiply out the answers we get and 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 it’s 6 and we’ll say what 2 numbers will add up to 6 and when those same 2 numbers are multiplied you’re going to get 8, well let’s just take about the factors of 8, the factors of 8 are 1 2 4 and 8, well 1 times 8 is 8, 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 zero. Now, if 2 expressions or2 numbers times each other equals zero that means that one of those 2 numbers or both of them must equal zero so we have to, now we could say that X plus 2 equals zero or and X plus 4 is equal to zero well this is 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 just try it with minus 2 and I will leave it minus 4 up to you. So minus 2 squared plus 6 times minus 2 plus 8, minus 2 squared, that’s a squared it’s 4 minus 12 6 times minus 2 plus 8 and sure enough that equals zero and if you did the same thing with negative 4 you’d also see that that works, and you might be saying well this is interesting this is an equation that has 2 solutions well if you think about it make sense because the graph of F of X is intersecting the X axis in 2 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 you want to figure out where it intersects the X axis we just set F of X is equal to zero now to swap the left and right, left and right sides of the equations and I get 2X squared plus 20X plus 50 equals zero, now with a little different this time from last time is here the coefficient on an X squared is actually 2 instead of the 1 and I like it to be a 1 so let’s divide the whole equation with the left and right side by 2 and I get X squared plus 10X plus 25 equals zero so all I did is multiply it ½ times this the same thing as dividing by 2 and then times ½ of course zero times ½ is zero. Now we’re ready to do what we did before and you might wanna pause it and try it yourself, we’re going to say X plus something times X plus something is equal to zero and those 2 not something’s 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 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 it turns out that both of these numbers are 5. And so you get X plus 5 equals zero or X plus 5 equals zero so you set it really right at once so you get X equals negative 5, so how do we think about this graphically I just told you al lot of these equations can intersect the X axis in 2 places but this one only has 1 solution, well this solution would look like this. If this is X equals negative 5 we’d have a parabola that just touches right there and then comes back up so instead of intersecting in 2 places it only intersects right there X equals negative 5 and now it’s an exercise just to prove to you that I’m not, I’m not teaching you incorrectly. Let’s multiply X plus times X plus 5just to show you it equals what 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 is, X squared plus 5X plus 5X 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 1 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 2 numbers when we add them they equal what’s the coefficient here it’s negative 1 so we can even say you can say to numbers A plus B equals minus 1and A times B will equal minus 30. Well, let's just think about what all the factors are of 30 there’s 1,2,3,5,6,10,15and 30 well something interesting is happening this time though since A times B is negative 30one of this numbers have to be negative they both can’t be negative cause if they are both negative then this would be positive 30 so A times B is negative 30 so its actually where going to have to say to this factors the difference between them should be negative 1 well if you look all of this all these numbers obviously when you pair them up they multiply up to 30 but the only one’s that have a difference of 1 is 5 and 6 and since it’s a negative 1 its going to be and I’m going to be very faster this and I’ll do more example problems XB X minus 6 times X plus 5 is equal to zero so how did I think about that negative 6 times 5 is negative 30negative 6 plus 5 is negative 1 so it works out and the more and more you do this practices I know it seems a little confusing right now it will make a lot more sense so you get X equals 6 or X negative 5 I think at this point you ready to try some solving quadratics by factoring and I’ll do couple more modules as soon as you get some more practice problems. Have fun!

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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 what exactly it’s going to look like, but it’s going to be a parabola and it’s going to intersect the X axis and then a couple of points here and here, and what we’re going to try to do is determine what those 2 point are.

So first of all when a function intersects the X axis that means F of X is equal to zero because this is the F of X axis somewhere to the Y axis, so here F of X is zero so in order to solve this equation we set F of X to zero and we get X squared plus 6X plus 8 is equal to zero.

Now this might look like you could solve it pretty easily but that X squared term kinda messes things up and you could try it out on, for yourself. So 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 we’ll still equal that, so that’s equal zero.

Now in this presentation I’m going to just show you the , the systematic or you could say the mechanical way of doing this, now I’m going to give you another presentation on why this works and you might wanna just multiply out the answers we get and 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 it’s 6 and we’ll say what 2 numbers will add up to 6 and when those same 2 numbers are multiplied you’re going to get 8, well let’s just take about the factors of 8, the factors of 8 are 1 2 4 and 8, well 1 times 8 is 8, 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 zero.

Now, if 2 expressions or2 numbers times each other equals zero that means that one of those 2 numbers or both of them must equal zero so we have to, now we could say that X plus 2 equals zero or and X plus 4 is equal to zero well this is 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 just try it with minus 2 and I will leave it minus 4 up to you.

So minus 2 squared plus 6 times minus 2 plus 8, minus 2 squared, that’s a squared it’s 4 minus 12 6 times minus 2 plus 8 and sure enough that equals zero and if you did the same thing with negative 4 you’d also see that that works, and you might be saying well this is interesting this is an equation that has 2 solutions well if you think about it make sense because the graph of F of X is intersecting the X axis in 2 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 you want to figure out where it intersects the X axis we just set F of X is equal to zero now to swap the left and right, left and right sides of the equations and I get 2X squared plus 20X plus 50 equals zero, now with a little different this time from last time is here the coefficient on an X squared is actually 2 instead of the 1 and I like it to be a 1 so let’s divide the whole equation with the left and right side by 2 and I get X squared plus 10X plus 25 equals zero so all I did is multiply it ½ times this the same thing as dividing by 2 and then times ½ of course zero times ½ is zero.

Now we’re ready to do what we did before and you might wanna pause it and try it yourself, we’re going to say X plus something times X plus something is equal to zero and those 2 not something’s 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 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 it turns out that both of these numbers are 5.

And so you get X plus 5 equals zero or X plus 5 equals zero so you set it really right at once so you get X equals negative 5, so how do we think about this graphically I just told you al lot of these equations can intersect the X axis in 2 places but this one only has 1 solution, well this solution would look like this.

If this is X equals negative 5 we’d have a parabola that just touches right there and then comes back up so instead of intersecting in 2 places it only intersects right there X equals negative 5 and now it’s an exercise just to prove to you that I’m not, I’m not teaching you incorrectly.

Let’s multiply X plus times X plus 5just to show you it equals what 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 is, X squared plus 5X plus 5X 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 1 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 2 numbers when we add them they equal what’s the coefficient here it’s negative 1 so we can even say you can say to numbers A plus B equals minus 1and A times B will equal minus 30.

Well, let's just think about what all the factors are of 30 there’s 1,2,3,5,6,10,15and 30 well something interesting is happening this time though since A times B is negative 30one of this numbers have to be negative they both can’t be negative cause if they are both negative then this would be positive 30 so A times B is negative 30 so its actually where going to have to say to this factors the difference between them should be negative 1 well if you look all of this all these numbers obviously when you pair them up they multiply up to 30 but the only one’s that have a difference of 1 is 5 and 6 and since it’s a negative 1 its going to be and I’m going to be very faster this and I’ll do more example problems XB

X minus 6 times X plus 5 is equal to zero so how did I think about that negative 6 times 5 is negative 30negative 6 plus 5 is negative 1 so it works out and the more and more you do this practices I know it seems a little confusing right now it will make a lot more sense so you get X equals 6 or X negative 5 I think at this point you ready to try some solving quadratics by factoring and I’ll do couple more modules as soon as you get some more practice problems. Have fun!

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