Algebra I
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CA Algebra I: Number Properties and Absolute Value
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CA Algebra I: Simplifying Expressions
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CA Algebra I: Simple Logical Arguments
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CA Algebra I: Graphing Inequalities
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CA Algebra I: Slope and Y-intercept
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CA Algebra I: Systems of Inequalities
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CA Algebra I: Simplifying Expressions
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CA Algebra I: Factoring Quadratics
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CA Algebra I: Completing the Square
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CA Algebra I: Quadratic Equation
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CA Algebra I: Quadratic Roots
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CA Algebra I: Rational Expressions 1
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CA Algebra I: Rational Expressions 2
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CA Algebra I: Word Problems
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CA Algebra I: More Word Problems
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CA Algebra I: Functions
CA Algebra I: Completing the Square 48-52, completing the square
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- We're on problem 48.
- It says if x squared is added to x, the sum is 42.
- So let's just write that out.
- If x squared is added to x, the sum is equal to 42.
- Which of the following could be the value of x?
- So essentially they just want us to solve this equation.
- So the easiest way to do it is to write it as a quadratic
- equaling 0 and then factoring it.
- So we could write this as x squared plus x minus 42 is
- equal to 0.
- And let's think.
- What two numbers when I add them equal 1, and when I
- multiply them equal minus 42?
- And the fact that when I multiply them equals minus 42
- tells me that one of them has to be positive and one of them
- has to be negative.
- That's the only way that when you multiply two numbers
- you're going to have a negative number.
- So one of them has to be positive, one of them has to
- be negative.
- And so when we're adding a positive and a negative,
- you're really finding the difference between the two.
- So the difference between the two numbers has to be 1 and
- their product has to be 42.
- And I noticed when I see 42 I immediately
- think, oh, 6 and 7.
- 6 times 7 is 42.
- And since when you add them you get a positive 1, 7's
- probably the positive one and minus 6 or 6 is probably the
- negative one.
- So let's try it out.
- x plus 7 times x minus 6 equal to 0.
- And right, 7 times minus 6 is minus 42.
- It's really 7x plus minus 6x is equal to positive x.
- Or you could think 7 plus minus 6 is equal to the
- coefficient on the x, which is 1.
- But anyway, that works.
- And you could multiply this out and try it out.
- And everything I'm saying, it's not some voodoo.
- The reason why I say that they have to add up to the 1 is
- because when you multiply this out, that's what
- builds up this term.
- This 7 times x plus the minus 6 times the other x.
- That's what builds this term when you multiply it out.
- This term comes from the x times the x.
- The minus 42 comes from the 7 times the minus 6.
- Anyway, now we're at this point.
- We say OK, well how do we get-- we have two things when
- you multiply them equal to 0.
- Well that means that one of them or both of them have to
- be equal to 0.
- So that means that x plus 7 is equal to 0, which would mean
- subtract 7 from both sides.
- Which means x is equal to minus 7.
- Or x minus 6 is equal to 0.
- Add 6 to both sides. x is equal to 6. x would
- be 6 or minus 7.
- And they have one of the choices there,
- which was choice A.
- Next problem.
- 49.
- What quantity should be added to both sides of this equation
- to complete the square?
- So when you complete the square you want the thing to
- just look like a-- you want whatever's on the left-hand
- side to look like a perfect square.
- And what do I mean by perfect square?
- So if I had x plus a squared, that's equal to x plus a
- times x plus a.
- And that's equal to x times x, x squared.
- x times that a, so that's plus ax.
- And now this a times this x.
- So that's another ax.
- Plus this a times that a.
- So plus a squared.
- And that's equal to x squared.
- Plus-- we have two of these now-- plus 2ax plus a squared.
- So essentially we want this, we want the left-hand side to
- have this form.
- So we say, this is a perfect square.
- We can say that's the same thing as x plus a squared.
- So let's think about how we can do it.
- If we have x squared minus 8x is equal to 5 and I put a
- space here for a reason because we want to add or
- subtract something here so it looks like a perfect square.
- So think about it.
- When we have this format, in order for this thing to be a
- perfect square, whatever this coefficient is right here,
- this term right here has to be half of this, squared.
- a squared is half of 2a squared.
- So if we took half of minus 8, that's minus 4.
- In this case, if we said 2a is equal to 8, a
- would be minus 4.
- And so minus 4 squared is what?
- It's plus 16.
- And this is an equation.
- So one you do to one side of an equation you have to do to
- the other side of the equation.
- So you have to say that that is also equal to.
- So you have to add 16 to both sides.
- Otherwise you're changing the equation.
- Now this, hopefully you recognize this as already a
- perfect square.
- I mean you could look at this pattern up here or you could
- say, OK, if I add minus 4 to itself twice I get minus 8.
- If I multiply it by itself I get 16.
- So this is x minus 4 squared.
- That's equal to 25.
- And actually, just if you're curious-- and we did this in
- the Khan Academy, we did a couple of videos on this--
- this is how you prove the quadratic equation.
- You essentially complete the square with arbitrary numbers
- a, b, and c, and you get the quadratic equation.
- You know, we show it in 10 minutes, so it's not this
- impossibly hard thing to understand.
- They just want to know, what do you add to both sides of
- this equation?
- What quantity should be added to both sides of this equation
- to complete the square?
- So the answer to this one was 16.
- But they just as equally could have said, solve it by
- completing the square.
- And you'd say oh, x minus 4 squared is equal to 25.
- So x minus 4 is equal to plus or minus 5.
- And then you could say, x is equal to plus or
- minus 5 plus 4.
- And then you could say, OK, which is 4 plus
- positive 5 is 9.
- 4 plus minus 5 is-- or, minus 1.
- Anyway, they didn't ask us that, so we don't have to
- spend too much time thinking about it.
- Let's see, we're on problem 50.
- Let me see, problem 50.
- I'll copy and paste 50 and 51.
- All right, what are the solutions for the quadratic
- equation x squared plus 6x is equal to 16?
- And the temptation here is really to kind of try to solve
- it the way you do a linear equation.
- I don't know, factor out an x and-- I don't know, do
- whatever else.
- But the important thing to recognize is this is a
- quadratic equation.
- And the easiest way to solve it is to put all the terms on
- one side and then get a 0 on the other side.
- And then either factor it or use the
- actual quadratic equation.
- Or complete the square, whatever you need to do.
- So let's subtract 16 from both sides.
- And you get x squared plus 6x minus 16 is equal to 0.
- I just subtracted 16 from both sides to get here.
- And before just jumping into the quadratic equation, let's
- see if we can factor it by inspection.
- So what two numbers, when I add them, equal 6-- and we
- want positive 6-- and when I multiply them equal minus 16?
- And once again, since it's a minus 16, if you multiply two
- numbers you get a negative number.
- They have to be different signs.
- One has to be positive and one has to be negative.
- And their difference will be 6 because one's positive and
- one's negative.
- So let me think about it.
- So if I had minus-- well, 8 and 2 is equal to 16.
- And they're 6 apart.
- So if I did plus 8 and minus 2-- right.
- Plus 8 and minus 2 is positive 6.
- So it's x plus 8 times x minus 2.
- And that really just takes a lot of practice.
- You say, OK, what two numbers?
- 16.
- OK.
- 8 and 2.
- Well they're going to have to be different signs.
- But I have a positive one here, so whichever number's
- larger is probably going to be the positive one.
- So positive 8 and minus 2.
- Yeah, when you add them up, they equal minus 6.
- Yeah, it works.
- So you set that equal to 0.
- And you say OK, this has to be equal to 0, or that has to be
- equal to 0.
- So x is either equal to minus 8.
- If you say x plus 8 is equal to 0 then subtract 8 from both
- sides, you get x is equal to minus 8.
- I shouldn't have skipped that step, but
- I'll do the step here.
- Or you could say x minus 2 is equal to 0.
- Add 2 to both sides, you get x is equal to 2.
- What x makes this term equal to 0?
- And you could look at it from inspection.
- So x could either be minus 8 or 2, and that is choice C.
- Problem 51.
- Leanne correctly solved the equation x squared plus 4x
- equals 6 by completing the square.
- Which equation is part of her solution?
- OK, so the same thing.
- x squared plus 4x.
- And when you complete the square you're going to add
- something here.
- So I'm going to leave a little blank.
- Is equal to 6.
- So what could I add here that makes this expression look
- like a perfect square?
- Well, got to the pattern that we did a
- couple of problems ago.
- Whatever's here should be the square of half of this.
- So 4-- well, half of that is 2.
- 2 squared is 4.
- So I should add 4 to that side.
- If I add 4 to that side, I have to add 4 to
- this side as well.
- And now this 2 plus 2 is equal to 4.
- 2 times 2 is equal to 4.
- So this is x plus 2 squared.
- And I really want you to get the intuition.
- Don't memorize the steps for completing the square.
- I want you to really understand why.
- This is the square of half of that.
- And we showed it in the beginning.
- Square a lot of binomials and see for yourself that that's
- always going to be the case.
- Anyway, so this is x plus 2 squared.
- That's going to be equal to-- 6 plus 4 is equal to 10.
- And that is choice B.
- I think we have time for one more.
- One more problem, problem 52.
- Copied it and now I have pasted it.
- Carter is solving this equation by factoring.
- Which expression could be one of his correct factors?
- Once again, I like to personally separate out the
- number that goes into all of them.
- And all of these are divisible by 5.
- And that just simplifies it in my head.
- So if I divide all of these by 5-- actually, I could just
- divide both sides of this equation by 5.
- 0 divided by 5 is 0.
- And then that left side divided by 5 becomes 2x
- squared minus 5x plus 3 is equal to 0.
- So if this is 2x squared here, so it's going to be two
- numbers when you multiply them equal 3 and when you-- so
- let's think about this a little bit.
- Actually, let me write it down here because I think I'll need
- more space.
- 2x squared minus 5x plus 3 is equal to 0.
- And I just divided both sides of the equation
- by 5 to get to this.
- So let's see what we can do here.
- So we have a 2x squared here and they already kind of
- hinted to us that we're going to have an integer solution,
- so we can factor this.
- So the intuition is that this is going to be 2x times-- you
- know, plus something.
- Plus a.
- Times-- well what times?
- Times probably x, right?
- 2x times x is 2x squared.
- Now that wouldn't be completely obvious if they
- didn't already tell us that we could factor this.
- You might have to use a
- quadratic equation or something.
- Actually, the quadratic equation wouldn't be something
- crazy to use here because you can just
- kind of plug and chug.
- But let's see if we can get the intuition.
- So it's going to be 2x plus something times x plus
- something else.
- If we were to multiply this out, you get 2x times x is 2x
- squared as it should.
- 2x times b is plus 2bx.
- a times x is plus ax.
- a times b is plus ab.
- And so let's see what we get.
- So plus 2b plus ax plus ab.
- 2x squared.
- OK, now can do pattern matching.
- This was our original thing.
- So 2 times b plus a has to be equal to-- this term is the
- same thing as this term right here.
- And that term has to be the same thing as
- that term right there.
- So first of all, I have a positive 3 here.
- So I'm multiplying two numbers to get a positive 3.
- So they either have to be both positive or both negative.
- And then the other interesting thing is we have-- when I take
- 2 times one of them plus the other one, I
- get a negative number.
- So the only way when you're dealing with negative numbers,
- and when you just multiply it times a positive and add them
- to each other you get another negative number, is if they're
- both negative.
- This told us they both have to be negative
- because this is positive.
- And then since when you add them without any negative
- signs you get a negative number, it tells you that that
- has to be negative as well.
- So let's see.
- Let's just try 3-- negative 3 and negative 1.
- If negative 3 and negative 1.
- So you're right.
- Yeah.
- If b is equal to minus 1 and a is equal to minus 3, then 2
- times minus 1 is minus 2.
- Minus 3.
- Right, so b is equal to minus 1 and a is equal to minus 3.
- This is a bit of an art form here.
- I mean there isn't like a plug and chug, very mechanical way
- of doing this.
- The quadratic equation is one, but this is the best way that,
- at least, I know how to do these without.
- So, we know what a and b is.
- So it's 2x-- a is minus 3.
- 2x minus 3 times x plus b. b is minus 1.
- So that's the factorization.
- So 2x minus 3 times x minus 1, which one?
- They have this one right here.
- 2x minus 3.
- And I'm all out of time.
- See you in the next video.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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