Algebra II
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California Standards Test: Algebra II
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California Standards Test: Algebra II (Graphing Inequalities
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CA Standards: Algebra II (Algebraic Division/Multiplication)
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CA Standards: Algebra II
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Algebra II: Simplifying Polynomials
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Algebra II: Imaginary and Complex Numbers
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Algebra II: Complex numbers and conjugates
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Algebra II: Quadratics and Shifts
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Examples: Graphing and interpreting quadratics
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Algebra ||: Conic Sections
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Algebra II: Circles and Logarithms
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Algebra II: Logarithms Exponential Growth
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Algebra II: Logarithms and more
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Algebra II: Functions, Combinatorics
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Algebra II: binomial Expansion and Combinatorics
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Algebra II: Binomial Expansions, Geometric Series Sum
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Algebra II: Functions and Probability
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Algebra II: Probability and Statistics
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Algebra II: Mean and Standard Deviation
Algebra II: Complex numbers and conjugates 27-31, complex numbers and conjugates
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- We're on problem 27.
- They tell us if i is equal to the square root of minus 1,
- then what is 4 times i-- that's an i-- times 6 times i?
- Well, multiplication is associative.
- You can switch the order around.
- So this is the same thing as 4 times 6 times i times i, or
- the same thing as 24 times i squared, right?
- If we square both sides of this, you get i squared is
- equal to negative 1.
- So this is 24 times negative 1.
- So this is minus 24, which is choice C.
- Next problem, 28.
- They want to know what an equivalent
- form of 3 plus i is.
- A lot of times, people don't like either when you have a
- square root in the denominator or they don't like it when you
- have an a complex number in the denominator.
- A complex number is just something that has part real
- and part imaginary.
- So the way you can do it with this case is you multiply
- times the conjugate of this number.
- I know that sounds like a fancy word, but all it means
- is you take the opposite of the imaginary part.
- So what's the conjugate?
- Remember, the only way you cannot change a fraction is if
- you multiply the numerator and the denominator by the same
- thing, because then you're multiplying by 1.
- So what's the conjugate of this?
- The conjugate of 3 plus i is 3 minus i.
- You have to do 3 minus i over 3 minus i.
- I'm doing that because when you multiply a complex number
- by its conjugate, you end up actually with a real number.
- Then you're going to end up with a complex number up here.
- A conjugate, all it means is you keep the real part the
- same, and you change the sign on the imaginary part.
- That's all that word means.
- All right, now let's figure out what this is equal to.
- 2 times 3, just same distributive
- properties as always.
- Let me do it in green.
- 2 times 3 is 6.
- 2 times minus i is minus 2i, all of that over
- 3 times 3 is 9.
- 3 times minus i is minus 3i.
- i times 3 is plus 3i.
- Then i times minus i, you can view that as minus i squared.
- Let's think about what this is equal to.
- Well, this plus 3i and the minus 3i cancel out.
- What's i squared?
- Well, this is equal to negative 1.
- So you end up with a minus of a negative 1.
- Minus a negative 1 is a plus 1.
- So you're left with 6 minus 2i over 9 plus 1 over 10.
- Actually, we can reduce this some.
- We can divide the top and the bottom by 2.
- So that becomes 3 minus i over 5.
- We just divided both the numerator and the
- denominator by 2.
- That is choice B.
- Problem 29, these go fast. No systems of
- equations and all that.
- What is the product of the complex numbers 3 plus
- i and 3 minus i?
- They just asked us that.
- That was in the last problem.
- When we multiplied 3 plus i times 3 minus
- i, what did we get?
- We got 10.
- We did this whole thing.
- 3 times 3 is 9.
- We did the whole thing.
- They're just repeating the same question.
- That equals 10.
- That's choice B.
- Rewind the video and see what I did here.
- 3 times 3 is 9 times minus 3i plus 3i minus i squared.
- These terms cancel out and you get a minus i squared, which
- is minus negative 1.
- So you have 9 plus 1, so that equals 10.
- So that is choice B.
- That's crazy.
- They just gave us the same problem twice in a row.
- All right.
- They're giving us more and more imaginary problems. But
- these are fun, especially if they're going to repeat the
- same question.
- Problem 30, if i is equal to the square root of negative
- 1-- well, I'll just keep writing it because they keep
- writing it.
- It's good as a reminder. i is equal to the square root of
- negative 1 and a and b are non-zero real numbers.
- What is 1 over a plus bi?
- Well, once again, if they kind of want us to rewrite this,
- they probably just want us to get rid of the i's in the
- denominator.
- So we multiply times the conjugate of
- this complex number.
- So we multiply times a plus bi, the conjugate over the
- conjugate, a plus bi.
- The conjugate isn't this whole thing.
- The conjugate is just a plus bi.
- Oh sorry, we should be multiplying times a minus bi.
- a minus bi over a minus bi.
- The conjugate of a plus bi, just so you get the
- terminology right, is just the a minus bi.
- But I'm doing a minus bi over a minus bi.
- So I'm not changing this fraction because this
- obviously is 1.
- x over x is the same thing as 1.
- So let's see what this is equal to.
- Well, the numerator just becomes a minus bi times 1,
- which is a minus bi.
- Then you have a times a, which is a squared.
- a times minus bi, so it's minus abi.
- Then you have bi times a, which is plus abi.
- I'll write it down here.
- They cancel out.
- Then you have bi times minus bi.
- So that's minus b squared i squared.
- These cancel out.
- So you're left with a minus bi over a squared
- minus b squared i squared.
- We know what i squared is.
- i squared is equal to negative 1.
- You square both sides of that and you get i squared is equal
- to negative 1.
- So if we multiply, this becomes a negative 1.
- But then this negative and this
- negative, they cancel out.
- You get a plus.
- So then we get a minus bi over a squared plus b squared.
- That is choice B.
- Next problem.
- All right, what are the solutions to the equation x
- squared plus 2x plus 2?
- All right, now, if you see an equation like this and you're
- like, OK, let's see, can I factor it?
- Oh, that is equal to 0.
- You say, OK, what number when I multiple them equal 2 and
- when I add them equal 2?
- Well, that's a hard one.
- Whenever you get stuck, the easiest thing to do is just
- use the quadratic equation, just as a reminder.
- If the equation is Ax squared plus Bx plus C-- maybe I
- should have done them all in in capital letters-- is equal
- to 0, the solution to this and the proof of this we've done
- in the Khan Academy, it comes from completing the square,
- but it's useful to memorize, is negative B.
- So the solutions are x are equal to negative B plus or
- minus the square root of B squared minus 4AC, all
- of that over 2A.
- So in this case, what is A?
- A is the coefficient on x squared.
- So A is equal to 1.
- What is B?
- B is the coefficient on x.
- So B is equal to 2.
- What is C?
- C is just a constant term.
- So C is equal to 2.
- We could just solve like that, just substitute these values.
- So you get x is equal to minus B, which is minus 2, plus or
- minus the square root of B squared, so that's 4,
- minus 4 minus A.
- So 4 minus 4 times A, 4 times 1, times 2, all of that--
- that's an ugly looking radical-- all of that over 2A,
- 2 times 1 is 2.
- So that equals, let's see, we get minus 2 plus or minus the
- square root of 4 minus-- this is 8, right?
- 4 times 1 times 2.
- 4 minus 8, which is minus 4, all of that over 2, which is
- equal to minus 2 plus or minus.
- So the square root of minus 4, we could rewrite that as the
- square root of 4 times minus 1.
- So that equals 2 plus or minus the square root of 4 times the
- square root of minus 1, all of that over 2, which is equal to
- minus 2 plus or minus-- the square root of 4 is 2 and the
- plus or minus is dealing with positive or negative 2-- times
- the square root of negative 1.
- Well, we learned that's i multiple times.
- The square root of negative 1 is i, all of that over 2.
- We can simplify this a little bit.
- I'll do it in a different color, just because I've run
- out of space at the bottom here.
- So this is equal to divide everything by 2, you get
- negative 1 plus or minus i.
- Let's see what the choices are.
- That's choice C, because negative 1 plus or minus i
- could also be written as negative 1 plus i as one value
- for x, and then the other one could be negative 1 minus i.
- They separate it out.
- They don't write plus or minus in the actual problem.
- I'm out of time.
- I'll see you in the next video. you in the next video
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?
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