Solving a quadratic by factoring factoring quadratics
Solving a quadratic by factoring
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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 six plus eight.
- 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 zero.
- So in order to solve this equation we set f of x to zero,
- and we get x squared plus sixx plus eight is equal to zero.
- 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 sixx plus eight, but
- this can be written as x plus something times x
- plus something.
- It will still equal that, except that's equal to zero.
- 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, six.
- And we say what two numbers will add up to six.
- And when those same two numbers are multiplied
- you're going to get eight.
- Well let's just think about the factors of eight.
- The factors of eight are one to four and eight.
- well one times eight is eight, but one plus eight is nine, so that doesn't work.
- two times four is eight, and two plus four is six.
- So that works.
- So we could just say x plus two and x plus four is equal to zero.
- 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 now we can say that x plus two equals zero, and x
- plus four is equal to zero.
- Well, this is a very simple equation.
- We subtract two from both sides and we get x equals negative two.
- And here we get x equals minus four.
- And if we substitute either of these into the
- original equation, we'll see that it works.
- Minus two-- so let's just try it with minus two and I'll leave
- minus four up to you --so minus two squared plus six times
- minus two plus eight.
- Minus two squared is four, minus twelve-- six times minus two --plus eight.
- And sure enough that equals zero.
- And if you did the same thing with negative four, 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 two x squared
- plus twentyx plus fifty.
- So if we want to figure out where it intersects the x-axis,
- we just set f of x equal to zero, and I'll just swap the left and
- right sides of the equation.
- And I get twox squared plus twentyx plus fifty equals zero.
- Now, what's a little different this time from last time, is
- here the coefficient on that x squared is actually a two instead
- of a one, and I like it to be a one.
- So let's divide the whole equation, both the left
- and right sides, by two.
- I get x squared plus tenx plus twenty-five equals zero.
- So all I did is I multiplied one / two times-- this is the same
- thing as dividing by two --times one / two.
- And of course zero times one / two is zero.
- 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 zero and those two somethings, they
- should add up to ten, and when you multiply them,
- they should be twenty-five.
- Let's think about the factors of twenty-five.
- You have one, five, and twenty-five.
- Well one times twenty-five is twenty-five.
- one plus twenty-five is twenty-six, not ten.
- five times five is twenty-five, and five plus five is ten, so five actually works.
- It actually turns out that both of these numbers are five.
- So you get x plus five equals zero or x plus five equals zero.
- So you just have to really write it once.
- So you get x equals negative five.
- 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 five, 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 five.
- And now as an exercise just to prove to you that I'm not
- teaching you incorrectly, let's multiply x plus five times x plus
- five 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
- five plus five times x plus five.
- x squared plus fivex plus fivex plus twenty-five.
- And that's x squared plus tenx plus twenty-five.
- 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 thirty is equal to zero.
- Once again, two numbers when we add them they equal-- whats the
- coefficient here, it's negative one.
- So we could say those two numbers are a plus b equals
- minus one and a times b will equal minus thirty.
- Well let's just think about what all the factors are of thirty.
- one, two, three, five, six, ten, fifteen, 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 they're both
- negative then this would be a positive thirty.
- a times b is negative thirty.
- So actually we're going to have 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 thirty.
- 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 with this and I'll do more example
- problems --this would be x minus six times x plus
- five is equal to zero.
- So how did I think about that?
- Negative six times five is negative thirty.
- Negative six plus five is negative one.
- 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 six or x equals negative five.
- 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.
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