Quadratic odds and ends
-
Solving a quadratic by factoring
-
CA Algebra I: Factoring Quadratics
-
Algebra II: Quadratics and Shifts
-
Examples: Graphing and interpreting quadratics
-
CA Algebra I: Completing the Square
-
Introduction to the quadratic equation
-
Quadratic Equation part 2
-
Quadratic Formula (proof)
-
CA Algebra I: Quadratic Equation
-
CA Algebra I: Quadratic Roots
Solving a quadratic by factoring factoring quadratics
⇐ Use this menu to view and help create subtitles for this video in many different languages.
You'll probably want to hide YouTube's captions if using these subtitles.
- 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.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
|
Have something that's not a question about this content? |
This discussion area is not meant for answering homework questions.
Discuss the site
For general discussions about Khan Academy, visit our Reddit discussion page.
Flag inappropriate posts
Here are posts to avoid making. If you do encounter them, flag them for attention from our Guardians.
abuse
- disrespectful or offensive
- an advertisement
not helpful
- low quality
- not about the video topic
- soliciting votes or seeking badges
- a homework question
- a duplicate answer
- repeatedly making the same post
wrong category
- a tip or feedback in Questions
- a question in Tips & Feedback
- an answer that should be its own question
about the site
Share a tip
Suggest a fix
Have something that's not a tip or feedback about this content?
This discussion area is not meant for answering homework questions.