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: Rational Expressions 1 63-65, simplifying rational expressions
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- We are on problem 63.
- The height of a triangle is 4 inches greater
- than twice its base.
- Let me this triangle in question.
- That's the triangle, that's its height, that's its base.
- Let's say this is the base.
- Let's call that b.
- So then the height, that's that, is 4 inches greater than
- twice its base.
- So 4 inches more than 2 times the base.
- Fair enough.
- The area of the triangle is 168 square inches.
- What is the base of the triangle?
- So what's the area of a triangle?
- It's the base times the height times 1/2.
- So area is equal to 1/2 base times height.
- Well they told us that the height is four more than twice
- the base, so we can just substitute that in.
- And then we'll have a just in terms of the base.
- Let me keep that there.
- So the area is equal to 1/2 the base times the height,
- which they told us is 4 more than twice the base.
- And now if we distribute the 1/2 b.
- 1/2 b times 4 is equal to 2b.
- And then 1/2 b times 2b, that's plus b squared.
- 1/2 time 2 is 1 and then b times b is b squared.
- Fair enough.
- And they tell us that the area is 168.
- OK, so we have area is 168, and that's equal to 2 times
- the base plus the base sqaured.
- Let's swap the sides just to make it look
- a little bit familiar.
- B squared plus 2 times the base is equal to 168.
- We could subtract 168 from both sides.
- You get b squared plus 2b minus 168 is equal to 0.
- And 168 is a bit of a strange number.
- I don't know exactly which numbers go into 168.
- Let's see, 4 isn't going to work.
- Does 12 go into it?
- Yeah, 12 goes into 168, and I think it
- goes into it 14 times.
- So that works.
- So you could use the quadratic equation to solve this.
- But I think that 12 times 14 is 168.
- And they have to be of different signs, since this is
- a negative.
- Only different signs when multiplied get
- you a negative number.
- And they're 2 apart.
- So that works out.
- So you get b plus 14 times b minus 12 is equal to 0.
- Let me just confirm and not waste your time.
- 14 times 12, I'm pretty sure I'm right.
- Yeah, 168.
- I just factored that out.
- So b could be equal to minus 14.
- Right, that would make that 0.
- Or b could be equal to 12, which would make that 0.
- Obviously if you can make one of these terms 0, you're
- making this whole thing 0.
- And the base, since we're dealing with distances or
- lengths, we can't have a negative length.
- So the base has to be 12 inches.
- Which is choice C in that question.
- Next question.
- Problem 64.
- All right.
- This is a good one.
- What is that thing reduced to lowest terms?
- So essentially what they want us to do is factor stuff out
- and factor it out and write it as a product of factors and
- then cancel out the ones.
- So let's see what we can do.
- Can we factor that top expression?
- Let's see.
- This is looks like a perfect square, right?
- Because if you have minus 2y squared is 4y squared.
- And minus 2y and minus 2y is minus 4y.
- So the top expression is x minus 2y squared.
- And I'll just write it twice.
- Because I'm guessing that one of them are going to have to
- be factored out.
- And once, the intuition is that if I had this, x squared
- minus 4x plus 4, you say OK, what two numbers when you
- multiply them equal four and when you add
- them equal minus 4?
- You say, minus 2 and you'd so that's equal to x minus
- 2 and x minus 2.
- The only difference between that and that is
- they put y's here.
- Y and a y squared.
- OK, so you could visualize this thing as x squared minus
- 4yx plus 4y squared.
- Or you could visualize this as x squared minus two times 2y x
- plus 2y squared.
- This isn't 4y squared.
- This is 4 y squared.
- Forget the parentheses.
- But I hope you can see that this thing up here is the same
- thing is this thing right here.
- And that's where I got the intuition that that's going to
- be x minus 2y times x minus 2y.
- You say, OK what two numbers when you add
- them are 2 times 2y?
- Well, obviously 2y.
- And what numbers when you squared them are 2y squared?
- Well 2y times 2y is 2y squared.
- So that's where you get the intuition that it's x minus 2y
- times x minus 2y.
- That's the numerator, let's see what we can do with the
- denominator.
- Well immediately, both factors are divisible by 3 and both
- terms are divisible by y.
- So let's factor those out.
- So this is equal to 3y.
- Now if you divide this term by 3y, all you're
- left with is an x.
- 3xy divided by 3y, you just have an x.
- And then if you divide minus 6y squared by 3y, minus six
- divided by 3 is minus 2, y squared divided by y is y.
- Let's see, if we assume that x does not equal 2y, because
- then this would be undefined, we can cancel these two out.
- And you're just left with x minus 2y over 3y.
- And that is choice B.
- Problem 65.
- They want us to do it again.
- OK, let's see.
- So this top expression, everything is divisible by 3,
- so let's rewrite it.
- Let's factor out a 3 there.
- So it'd be 3 times to 2x squared, plus 7x-- or 21
- divided by 3-- Plus 3.
- Let's see, what's the bottom one?
- Well this fits a pattern.
- A squared minus b squared, right?
- This is something squared.
- 4x squared is 2x squared.
- And then 1 is obviously 1 squared.
- So we can write this as a plus b times a minus b.
- And that would be 2x plus 1, times 2x minus 1.
- 2x squared is 4x squared, and 1 squared is 1.
- So let's see, we factored that top part.
- And I'm guessing it's going to cancel out with something in
- the denominator.
- So let's see what we can get.
- Let me do it on the side here.
- So if I have to factor 2x squared plus 7x plus 3, it's
- going to be 2x plus something times x plus something.
- Because 2x times the x is 2x squared.
- And the b times the a is going to be equal to 3.
- So I'm guessing one of them is going to be a 3 and one of
- them is going to be 1.
- And I could already guess, if we could do it a little bit
- more systematically than how it's going, so
- let's multiply it out.
- 2x times x, 2x squared.
- 2x times a, plus 2ax.
- B times x, plus bx.
- B times a, plus ba.
- So you get 2x squared, then merge these terms, add the
- coefficients.
- Plus 2a plus bx, that's those two terms. Plus ba is equal to
- this thing up here.
- I'll write it right below it so we can compare terms. It's
- equal to 2x squared plus 7x plus 3.
- So what could a and b equal?
- Let's see, 2 times a plus b is equal to-- so if a is 3 and
- this is 1, 2 times 3 is 6 plus 1 is 7, works.
- And then 3 times 1 is 7.
- So b is 1 and a is 3.
- And you could multiply that out if you're not
- comfortable with it.
- But anyway, we could rewrite this top thing as 3 times--
- and we did the factoring here, we figured out that b and a
- are 2x plus 1, times x plus 3.
- All of that over 2x plus 1, times 2x minus 1.
- And we know we're on the right track because we've got
- something to cancel out.
- And sure enough, we cancel those out.
- Assuming that this never equals 0.
- This x does not equal minus 1/2, because that would make
- that equal 0.
- And we get 3 times x plus 3 over 2x minus 1.
- And that is choice B.
- Let me see the next problem, see if we have time
- to do it right now.
- Oh, it's another one of these.
- I'll wait for the next video, see you soon.
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