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 zero.
Because this is 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 messes things up and you could
try it out for yourself.
So we're going to do is factor this.
And we're going to say that x squared plus 6x plus 8, but
this can be written as x plus something times x
plus something.
It will still equal that, except that's equal to 0.
Now in this presentation, I'm going to just show you the
systematic or you could say the mechanical way of doing this.
I'm going to give you another presentation on why this works.
And you might want to just multiply out the answers we
get in and multiply out the expressions and
see why it works.
And the message we're going to use is, we look at the
coefficient on this x term, 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 one to 4 and 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
0, that means that one of those two numbers or both of
them must equal 0.
So now we can say that x plus 2 equals 0, 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'll leave
minus 4 up to you --so minus 2 squared plus 6 times
minus 2 plus 8.
Minus 2 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 works.
And you might be saying, wow, this is interesting.
This is an equation that has two solutions.
Well, if you think about it, 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 2 x squared
plus 20x plus 50.
So if we want to figure out where it intersects the x-axis,
we just set f of x equal to 0, and I'll just swap the left and
right sides of the equation.
And I get 2x squared plus 20x plus 50 equals 0.
Now, what's a little different this time from last time, is
here the coefficient on that x squared is actually a 2 instead
of a 1, and I like it to be a 1.
So let's divide the whole equation, both the left
and right sides, by 2.
I get x squared plus 10x plus 25 equals 0.
So all I did is I multiplied 1/2 times-- this is the same
thing as dividing by 2 --times 1/2.
And of course 0 times 1/2 is 0.
Now we are 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 0 and those two somethings, 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.
1 plus 25 is 26, not 10.
5 times 5 is 25, and 5 plus 5 is 10, so 5 actually works.
It actually turns out that both of these numbers are 5.
So you get x plus 5 equals 0 or x plus 5 equals 0.
So you just have to really write it once.
So you get x equals negative 5.
So how do you think about this graphically?
I just told you that these equations can intersect the
x-axis in two places, but this only has one solution.
Well, this solution would look like.
If this is x equals negative 5, we'd have a parabola that just
touches right there, and then comes back up.
And instead of intersecting in two places it only
intersects right there at x equals negative 5.
And now as an exercise just to prove to you that I'm not
teaching you incorrectly, 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 is x times x plus
5 plus 5 times x plus 5.
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 one more problem.
And this one I am just going to cut to the chase.
Let's just solve x squared minus x minus 30 is equal to 0.
Once again, two numbers when we add them they equal-- whats the
coefficient here, it's negative 1.
So we could say those two numbers are a plus b equals
minus 1 and a times b will equal minus 30.
Well let's just think about what all the factors are of 30.
1, 2, 3, 5, 6, 10, 15, 30.
Well, something interesting is happening this time though.
Since a times b is negative 30, one of these numbers
have to be negative.
They both can't be negative, because if they're both
negative then this would be a positive 30.
a times b is negative 30.
So actually we're going to have to say, two of these factors,
the difference between them should be negative 1.
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 1 is 5 and 6.
And since it's a negative 1, it's going to be-- and I know
I'm going very fast with this and I'll do more example
problems --this would 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 solving
quadratics by factoring and I'll do a couple more modules
as soon as you get some more practice problems.
Have fun.