Example 3: Completing the square U10_L1_T2_we3 Completing the Square 3
Example 3: Completing the square
⇐ 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.
- Use completing the square to write the quadratic equation y
- is equal to negative 3x squared, plus 24x, minus 27 in
- vertex form, and then identify the vertex.
- So we'll see what vertex form is, but we essentially
- complete the square, and we generate the function, or we
- algebraically manipulate it so it's in the form y is equal to
- A times x minus B squared, plus C.
- We want to get the equation into this form right here.
- This is vertex form right there.
- And once you have it in vertex form, you'll see that you can
- identify the x value of the vertex as what value will make
- this expression equal to 0.
- So in this case it would be B.
- And the y value of the vertex, if this is equal to 0, then
- the y value is just going to be C.
- And we're going to see that.
- We're going to understand why that is the vertex, why this
- vertex form is useful.
- So let's try to manipulate this equation to get
- it into that form.
- So if we just rewrite it, the first thing that immediately
- jumps out at me, at least, is that all of these numbers are
- divisible by negative 3.
- And I just always find it easier to manipulate an
- equation if I have a 1 coefficient out in front of
- the x squared.
- So let's just factor out a negative 3
- right from the get-go.
- So we can rewrite this as y is equal to negative 3 times x
- squared, minus 8x-- 24 divided by negative 3 is
- negative 8-- plus 9.
- Negative 27 divided by negative 3 is positive 9.
- Let me actually write the positive 9 out here.
- You're going to see in a second why I'm doing that.
- Now, we want to be able to express part of this
- expression as a perfect square.
- That's what vertex form does for us.
- We want to be able to express part of this expression as a
- perfect square.
- Now how can we do that?
- Well, we have an x squared minus 8x.
- So if we had a positive 16 here-- because, well, just
- think about it this way, if we had negative 8, you divide it
- by 2, you get negative 4.
- You square that, it's positive 16.
- So if you had a positive 16 here, this would
- be a perfect square.
- This would be x minus 4 squared.
- But you can't just willy-nilly add a 16 there, you would
- either have to add a similar amount to the other side, and
- you would have to scale it by the negative 3 and all of
- that, or, you can just subtract a 16 right here.
- I haven't changed the expression.
- I'm adding a 16, subtracting a 16.
- I've added a 0.
- I haven't it changed it.
- But what it allows me to do is express this part of the
- equation as a perfect square.
- That right there is x minus 4 squared.
- And if you're confused, how did I know it was 16?
- Just think, I took negative 8, I divided by 2, I
- got negative 4.
- And I squared negative 4.
- This is negative 4 squared right there.
- And then I have to subtract that same amount so I don't
- change the equation.
- So that part is x minus 4 squared.
- And then we still have this negative 3 hanging out there.
- And then we have negative 16 plus 9, which is negative 7.
- So we're almost there.
- We have y equal to negative 3 times this whole thing, not
- quite there.
- To get it there, we just multiply negative 3.
- We distribute the negative 3 on to both of these terms. So
- we get y is equal to negative 3 times x, minus 4 squared.
- And negative 3 times negative 7 is positive 21.
- So we have it in our vertex form, we're done with that.
- And if you want to think about what the vertex is, I told you
- how to do it.
- You say, well, what's the x value that makes
- this equal to 0?
- Well, in order for this term to be 0, x minus 4 has to be
- equal to 0.
- x minus 4 has to be equal to 0, or add 4 to both sides. x
- has to be equal to 4.
- And if x is equal to 4, this is 0, this whole thing becomes
- 0, then y is equal to 21.
- So the vertex of this parabola-- I'll just do a
- quick graph right here-- the vertex of this parabola occurs
- at the point 4, 21.
- So I'll draw it like this.
- Occurs at the point.
- If this is the point 4, if this right here is the-- so
- this is the y-axis, that's the x-axis-- so this is
- the point 4, 21.
- Now, that's either going to be the minimum or the maximum
- point in our parabola, and to think about whether it's the
- minimum or maximum point, think about what happens.
- Let's explore this equation a little bit.
- This thing, this x minus 4 squared is always greater than
- or equal to 0.
- At worst it could be 0, but you're taking a square, so
- it's going to be a non-negative number.
- But when you take a non-negative number, and then
- you multiply it by negative 3, that guarantees that this
- whole thing is going to be less than or equal to 0.
- So the best, the highest, value that this function can
- attain, is when this expression right
- here is equal to 0.
- And this expression is equal to 0 when x is equal
- to 4 and y is 21.
- So this is the highest value that the function can attain.
- It can only go down from there.
- Because if you shift the x around 4, then this expression
- right here will become, well, it'll become non-zero.
- When you square it, it'll become positive.
- When you multiply it by negative 3,
- it'll become negative.
- So you're going to take a negative number plus 21, it'll
- be less than 21, so your parabola is going
- to look like this.
- Your parabola is going to look like that.
- And that's why vertex form is useful.
- You break it up into the part of the equation that changes
- in value, and say, well, what's its
- maximum value attained?
- That's the vertex.
- That happens when x is equal to 4.
- And you know its y value.
- And because you have a negative coefficient out here
- that's a negative 3, you know that it's going to be a
- downward opening graph.
- If that was a positive 3, then this thing would be, at
- minimum, 0 and it would be an upward opening graph.
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.
Share a tip
When naming a variable, it is okay to use most letters, but some are reserved, like 'e', which represents the value 2.7831...
Have something that's not a tip or feedback 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.
- disrespectful or offensive
- an advertisement
- low quality
- not about the video topic
- soliciting votes or seeking badges
- a homework question
- a duplicate answer
- repeatedly making the same post
- a tip or feedback in Questions
- a question in Tips & Feedback
- an answer that should be its own question
about the site