Completing the square
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Solving Quadratic Equations by Square Roots
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Example: Solving simple quadratic
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Solving quadratics by taking the square root
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Solving Quadratic Equations by Completing the Square
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Completing the square (old school)
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Example: Completing perfect square trinomials
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Example 1: Completing the square
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Example 2: Completing the square
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Example 3: Completing the square
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Example 4: Completing the square
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Example 5: Completing the square
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Completing the square 1
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Completing the square 2
Example 3: Completing the square U10_L1_T2_we3 Completing the Square 3
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- 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.
- Right?
- 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.
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