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Welcome to solving a quadratic by factoring.

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Let's start doing some problems.

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So, let's say I had a function f of x is equal to x

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squared plus 6x plus 8.

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Now if I were to graph f of x, the graph is going to

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look something like this.

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I don't know exactly what it's going to look like, but it's

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going to be a parabola and it's going to intersect the x-axis

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at a couple of points, here and here.

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And what we're going to try to do is determine what

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those two points are.

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So first of all, when a function intersects the

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x-axis, that means f of x is equal to zero.

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Because this is f of x-axis, similar to the y-axis.

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So here f of x is 0.

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So in order to solve this equation we set f of x to 0,

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and we get x squared plus 6x plus 8 is equal to 0.

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Now this might look like you could solve it pretty easily,

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but that x squared term messes things up and you could

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try it out for yourself.

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So we're going to do is factor this.

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And we're going to say that x squared plus 6x plus 8, but

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this can be written as x plus something times x

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plus something.

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It will still equal that, except that's equal to 0.

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Now in this presentation, I'm going to just show you the

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systematic or you could say the mechanical way of doing this.

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I'm going to give you another presentation on why this works.

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And you might want to just multiply out the answers we

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get in and multiply out the expressions and

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see why it works.

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And the message we're going to use is, we look at the

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coefficient on this x term, 6.

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And we say what two numbers will add up to 6.

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And when those same two numbers are multiplied

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you're going to get 8.

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Well let's just think about the factors of 8.

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The factors of 8 are one to 4 and 8.

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well 1 times 8 is 8, but 1 plus 8 is 9, so that doesn't work.

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2 times 4 is 8, and 2 plus 4 is 6.

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So that works.

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So we could just say x plus 2 and x plus 4 is equal to 0.

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Now if two expressions or two numbers times each other equals

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0, that means that one of those two numbers or both of

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them must equal 0.

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So now we can say that x plus 2 equals 0, and x

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plus 4 is equal to zero.

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Well, this is a very simple equation.

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We subtract 2 from both sides and we get x equals negative 2.

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And here we get x equals minus 4.

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And if we substitute either of these into the

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original equation, we'll see that it works.

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Minus 2-- so let's just try it with minus 2 and I'll leave

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minus 4 up to you --so minus 2 squared plus 6 times

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minus 2 plus 8.

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Minus 2 squared is 4, minus 12-- 6 times minus 2 --plus 8.

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And sure enough that equals 0.

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And if you did the same thing with negative 4, you'd

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also see that works.

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And you might be saying, wow, this is interesting.

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This is an equation that has two solutions.

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Well, if you think about it, it makes sense because the graph

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of f of x is intersecting the x-axis in two different places.

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Let's do another problem.

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Let's say I had f of x is equal to 2 x squared

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plus 20x plus 50.

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So if we want to figure out where it intersects the x-axis,

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we just set f of x equal to 0, and I'll just swap the left and

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right sides of the equation.

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And I get 2x squared plus 20x plus 50 equals 0.

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Now, what's a little different this time from last time, is

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here the coefficient on that x squared is actually a 2 instead

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of a 1, and I like it to be a 1.

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So let's divide the whole equation, both the left

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and right sides, by 2.

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I get x squared plus 10x plus 25 equals 0.

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So all I did is I multiplied 1/2 times-- this is the same

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thing as dividing by 2 --times 1/2.

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And of course 0 times 1/2 is 0.

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Now we are ready to do what we did before, and you

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might want to pause it and try it yourself.

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We're going to say x plus something times x plus

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something is equal to 0 and those two somethings, they

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should add up to 10, and when you multiply them,

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they should be 25.

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Let's think about the factors of 25.

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You have 1, 5, and 25.

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Well 1 times 25 is 25.

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1 plus 25 is 26, not 10.

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5 times 5 is 25, and 5 plus 5 is 10, so 5 actually works.

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It actually turns out that both of these numbers are 5.

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So you get x plus 5 equals 0 or x plus 5 equals 0.

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So you just have to really write it once.

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So you get x equals negative 5.

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So how do you think about this graphically?

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I just told you that these equations can intersect the

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x-axis in two places, but this only has one solution.

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Well, this solution would look like.

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If this is x equals negative 5, we'd have a parabola that just

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touches right there, and then comes back up.

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And instead of intersecting in two places it only

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intersects right there at x equals negative 5.

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And now as an exercise just to prove to you that I'm not

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teaching you incorrectly, let's multiply x plus 5 times x plus

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5 just to show you that it equals what it should equal.

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So we just say that this is the same thing is x times x plus

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5 plus 5 times x plus 5.

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x squared plus 5x plus 5x plus 25.

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And that's x squared plus 10x plus 25.

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So, it equals what we said it should equal.

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And I'm going to once again do another module where I explain

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this a little bit more.

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Let's do one more problem.

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And this one I am just going to cut to the chase.

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Let's just solve x squared minus x minus 30 is equal to 0.

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Once again, two numbers when we add them they equal-- whats the

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coefficient here, it's negative 1.

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So we could say those two numbers are a plus b equals

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minus 1 and a times b will equal minus 30.

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Well let's just think about what all the factors are of 30.

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1, 2, 3, 5, 6, 10, 15, 30.

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Well, something interesting is happening this time though.

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Since a times b is negative 30, one of these numbers

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have to be negative.

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They both can't be negative, because if they're both

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negative then this would be a positive 30.

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a times b is negative 30.

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So actually we're going to have to say, two of these factors,

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the difference between them should be negative 1.

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Well, if we look at all of these, all these numbers

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obviously when you pair them up, they multiply out to 30.

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But the only ones that have a difference of 1 is 5 and 6.

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And since it's a negative 1, it's going to be-- and I know

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I'm going very fast with this and I'll do more example

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problems --this would be x minus 6 times x plus

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5 is equal to 0.

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So how did I think about that?

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Negative 6 times 5 is negative 30.

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Negative 6 plus 5 is negative 1.

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So it works out.

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And the more and more you do these practices-- I know it

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seems a little confusing right now --it'll make

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a lot more sense.

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So you get x equals 6 or x equals negative 5.

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I think at this point you're ready to try some solving

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quadratics by factoring and I'll do a couple more modules

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as soon as you get some more practice problems.

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Have fun.

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