CA Algebra I: Quadratic Equation 53-57, Quadratic Equation
CA Algebra I: Quadratic Equation
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- We're on problem 53.
- It says Toni is solving this equation by completing the
- square. ax squared plus bx plus c is equal to 0, where a
- is greater than 0.
- So this is just a traditional quadratic right here.
- And let's see what they did.
- First, he subtracted c from both sides and he got ax
- squared plus bx is equal to minus c.
- OK, that's fair enough.
- And then let's see.
- He divided both sides by a.
- Right, that's fair enough.
- He got minus c/a.
- Which step should be Step 3 in the solution?
- So he's completing the square.
- So essentially, he wants this to become a perfect square.
- So let's see how we can do that.
- So we have x squared plus b/a x-- and I'm going to leave a
- little space here-- is equal to minus c/a.
- So for this to be a perfect square we have to add
- something here, we have to add a number.
- And we learned from several videos in the past and we kind
- of pseudo-proved it.
- And actually, I have several videos I do solely on
- completing the square.
- You essentially have to add whatever number this is, add
- half of it squared.
- And if that doesn't make sense to you, watch the Khan Academy
- video on completing the square.
- But what is half of b/a?
- Well it's b over 2a.
- So 1/2 times b/a is equal to b over 2a.
- And then, we want to add this squared.
- So let's add that to both sides of this equation.
- So we're left with x squared plus b/a x.
- And we want to add this squared.
- Plus b over 2a squared is equal to minus c/a.
- Anything you add to one side of the equation, you have to
- add to the other.
- So we have to add that to both sides.
- Plus b over 2a squared.
- And let's see if we've solved the problem so
- far, what they want.
- X, b over 2-- right.
- This is exactly what we did. x squared plus b/a plus b over
- 2a squared, and they add it to both sides of the equation.
- So D is the right answer.
- Now if you find that a little confusing or if it wasn't
- intuitive for you, I don't want you to
- memorize the steps.
- Watch the Khan Academy video on completing the square.
- Next problem, 56.
- No, 54.
- All right, this is another one that should be cut and pasted.
- All right, four steps to derive the quadratic formula
- are shown below.
- I said in previous videos that you can derive the quadratic
- formula by completing the square.
- And we actually do that in another video.
- I don't want to give too much of a plug for other videos,
- but let's see what they want to do.
- What is the correct order of these steps?
- So the first thing you want to start off with is just a
- quadratic equation.
- And this one is the first step.
- This is where we started off with in the last problem.
- Then what you want to do is add 1/2 of this squared to
- both sides.
- So b over 2a squared you want to add to both sides, and
- that's what they did here.
- So our order is I.
- And then you want to do IV.
- That's what we did in the last problem.
- We did IV.
- And then from here, you know that this expression right
- here is going to be equal to x plus b over 2a squared.
- And once again, watch soon. the completing the squared
- video if that didn't make sense.
- But the whole reason why you added this here is so that you
- know that, OK, what two numbers, when I multiply them
- equal b over 2a squared, and when I add them equal b/a?
- Well that's obviously, b over 2a.
- If you add it twice you're going to get b over a.
- If you square it, you're going to get this whole expression.
- So you say, oh, this is just x plus b over 2a squared and you
- get that there.
- And then, is equal to-- and then they just
- simplify this fraction.
- They found a common denominator and all the rest.
- And so the next step is Step II.
- And then all you have left is Step III.
- And you've pretty much derived the quadratic equation.
- So I, IV, II, III.
- That's choice A.
- Problem 55.
- Which of the solutions-- OK, I'll put all
- of the choices down.
- So which is one of the solutions to the equation?
- So immediately when you see all of the choices, they have
- these square roots and all that.
- This isn't something that you would factor.
- You would use a quadratic equation here.
- So let's do that.
- So the quadratic equation is, so if this is Ax squared plus
- Bx plus C is equal to 0.
- The quadratic equation is minus b.
- Well they do it lowercase.
- Plus or minus the square root of b squared minus 4ac, all of
- that over 2a.
- And this is just derived from completing the square with
- this, but we do that in another video.
- And so let's substitute it in.
- What is b?
- b is minus 1, right?
- So minus minus 1, that's a positive 1.
- Plus or minus the square root of b squared.
- Minus 1 squared is 1.
- Minus 4 times a.
- a is 2.
- Times 2.
- Times c.
- c is minus 4.
- So times minus 4.
- All of that over 2a.
- a is 2, so 2 times a is 4.
- So that becomes 1 plus or minus the square root.
- So we have a 1.
- So we have minus 4 times a 2 times a minus 4.
- That's the same thing as a plus 4 times 2 times a plus 4.
- Let's just take that minus out.
- So it's plus.
- There's no minus here.
- So let's see, 4 times 2 is 8.
- Times 4 is 32.
- Plus 1 is 33.
- All of that over 4.
- Let's see, we're not quite there yet.
- Well they say, which is one of the solutions to the equation?
- So let's see.
- If we wanted to simplify this out a-- well,
- this is right here.
- Because we have 1 plus or minus the square
- root of 33 over 4.
- Well they wrote just one of them.
- They wrote just the plus.
- So C is one of the solutions.
- The other one would have been if you had a minus sign here.
- Anyway, next problem.
- And this is another one I need to cut and paste.
- It says, which statement best explains why there's no real
- solution to the quadratic equation?
- OK, so I already have a guess of why this
- won't have a solution.
- But in general-- well, let's try the quadratic equation.
- Before even looking at this problem,
- let's get an intuition.
- It's negative b plus or minus you the square root of b
- squared minus 4ac, all of that over 2a.
- My question is to you, when does this not make any sense?
- Well you know, this'll work for any b, any 2a.
- But when does the square root sign really fall apart, at
- least when we're dealing with real numbers,
- and that's a clue?
- Well, it's when you have a negative number under here.
- If you end up with a negative number under the square root
- sign, at least if we haven't learned imaginary numbers yet,
- you don't know what to do.
- There's no real solution to the quadratic equation.
- So if b squared minus 4ac is less than
- 0, you're in trouble.
- There's no real solution.
- You can't take a square root of a negative sign if you're
- doing with real numbers.
- So that's probably going to be the problem here.
- So let's see what b squared minus 4ac is.
- You have b is 1.
- So 1 minus 4 times a.
- a is 2.
- 2 times c is 7.
- And sure enough, 1 times 4 times 2 times 7 is going to be
- less than 0.
- So let's just see what they have here.
- Right, the value of 1 squared-- oh, right.
- It's b squared.
- Well 1 squared, same thing as 1.
- 1 squared minus 4 times 2 times 7,
- sure enough is negative.
- So that's why we don't have a real
- solution to this equation.
- Next problem.
- I'm actually out of space.
- OK, they want to know the solution set to
- this quadratic equation.
- I'll just copy and paste.
- So that's essentially the set of the x's that
- satisfy this equation.
- And obviously, for any x that you put in this, the left-hand
- side is going to be equal to 0.
- So what x's are valid?
- And they just want us to apply the quadratic equation.
- So we've written it a couple of times, but let's just do it
- straight up.
- So it's negative b.
- b is 2.
- So it's negative 2 plus or minus the
- square root of b squared.
- Well that's 2 squared.
- Minus 4 times a.
- a is 8.
- Times c, which is 1.
- All of that over 2 times a.
- So 2 times 8, which is equal to minus 2 plus or minus the
- square root of 4-- let's see.
- Did I write this down?
- Negative b plus or minus the square root of b squared minus
- 4 times a times c.
- So you get 4 minus 32.
- That's why I was double checking to see if I did this
- right because I'm going to get a negative number here.
- All of that over 16.
- And so we're going to end up with the same conundrum we had
- in the last. 4 minus 32, we're going to end with minus 2 plus
- or minus the square root of minus 28 over 16.
- And if we're dealing with real numbers, I mean there's no
- real solution here.
- And at first I was worried.
- I thought I made a careless mistake or there was an error
- in the problem.
- But then I look at the choices.
- They have choice D.
- And I'll copy and paste choice D here.
- Choice D.
- No real solution.
- So that's the answer, because you can't take a square root
- of a negative number and stay in the set of real numbers.
- Let's see, do I have time for another one?
- I'm over the 10 minutes.
- I'll wait for the next video.
- See you soon.
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