Main content

## SAT

### Course: SAT > Unit 10

Lesson 2: Passport to advanced mathematics- Solving quadratic equations — Basic example
- Solving quadratic equations — Harder example
- Interpreting nonlinear expressions — Basic example
- Interpreting nonlinear expressions — Harder example
- Quadratic and exponential word problems — Basic example
- Quadratic and exponential word problems — Harder example
- Manipulating quadratic and exponential expressions — Basic example
- Manipulating quadratic and exponential expressions — Harder example
- Radicals and rational exponents — Basic example
- Radicals and rational exponents — Harder example
- Radical and rational equations — Basic example
- Radical and rational equations — Harder example
- Operations with rational expressions — Basic example
- Operations with rational expressions — Harder example
- Operations with polynomials — Basic example
- Operations with polynomials — Harder example
- Polynomial factors and graphs — Basic example
- Polynomial factors and graphs — Harder example
- Nonlinear equation graphs — Basic example
- Nonlinear equation graphs — Harder example
- Linear and quadratic systems — Basic example
- Linear and quadratic systems — Harder example
- Structure in expressions — Basic example
- Structure in expressions — Harder example
- Isolating quantities — Basic example
- Isolating quantities — Harder example
- Function notation — Basic example
- Function notation — Harder example

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Manipulating quadratic and exponential expressions — Harder example

Watch Sal work through a harder Manipulating quadratic and exponential expressions problem.

## Want to join the conversation?

- Can someone help me with this beginning to end?. i dont understand ho to do any of this.(38 votes)
- It's confusing when he says "nonpositive" or "nonnegative" why wouldn't just say negative and positive?(21 votes)
- From where in khan academy i can get explanation about w= -b/2a which sal used at 6.40(7 votes)
- Check it out on Quadratic Functions. It is derived from the Completing Square Method.

Anyway, the sum of roots of a quadratic equation = -b/a

Vertex lies on axis of symmetry, midway between the two roots,

So, abscissa of vertex = -b/2a

As far as I remember, it was covered on the Parabola section in Quadratic functions tutorial.(13 votes)

- what should do i am getting 1400 in collegeboard exams but I am getting only 1200 in kaplan and barron tests.(9 votes)
- I recommend to do as much as prep books/questions as you can. Especially understand the 'syllabus' and what's required of you to learn. Finish all of Khan Academy's exercises, correct your errors, learn time conservation methods and practice!! Keep on practicing!(12 votes)

- Why did it have to be 50?(10 votes)
- How'd he get 100w at 0.51?? 50 squared is 2500.(1 vote)
- What he did was he expanded (w-50)^2. This is how you do it:

(w-50)(w-50)

w^2-50w-50w+2500

w^2-100w+2500

He didn't show the steps and combined the -50w and -50w. He then did 50^2 which is 2500.

Hope this helps!(20 votes)

- Another way to get this is that, this is the "completing square" form, therefore its easier to get the maximum point. Like he said, vertex form.(5 votes)
- in00:47he's getting 100 out of nowhere. Can anyone pls explain!(3 votes)
- There, Sal is applying a property of squared binomials:

(a + b)^2 = a^2 + 2*a*b + b^2

This essentially takes a step out of foiling the two binomials together. -100w is the middle term, (2)(-50)(w).(3 votes)

- thank god for differentiation man.(4 votes)
- why does 50 have to be used? Also, since all of the answer options are equal to the original one could you use any of the numbers like 20 or 10?(3 votes)
- The "50" is obtained by taking the original equation and completing the square. If you don't know what completing the square is, Khan Academy has some videos on it.(2 votes)

## Video transcript

- [Instructor] Dina
purchased 200 feet of fencing to make a rectangular
play area for her dogs. The possible area A, is
given by the equation below where w is the
width of the play area. So, they do the work for
us, where they tell us what area as a function
of w is going to be. Which of the following
equivalent expressions displays, as a constant or a coefficient, the value of the width for
which the area is a maximum? So that's an interesting
way that they phrased it, but they're saying that these are all equivalent expressions
to this thing up here. And you could verify that
they're equivalent expressions, or we could take, for example, this first one right over here, this would be the same thing as negative, let's see w minus 50 squared is w squared minus two times 50 times w,
so minus 100 w plus 2500. And then, you have plus 2500 out here. Now, this is gonna be the same thing as negative w squared plus, I'm just distributing the negative sign, plus 100 w minus 2500 plus 2500, which these two things
cancel each other out, and you're left with this
expression right over here. Negative w squared, negative
w squared plus a hundred w. This would be true for all
of these, so these are all just algebraic manipulations
of this original thing. Now, what they're asking us, or one way to think about it is, which of these show the maximum value? Which of these forms of
the expression make it easy for us to find what the
maximum value's going to be? Well, let's look at this first one. This first one, let's think
about how this expression works. If this, this part right over here, w minus 50 squared that's
always going to be non-negative 'cause if you take the
square of something, it's going to be non-negative. But then, if you then take the negative of it, it's going to be non-positive. It's going to be non-positive, so this part over here
is going to be less than, or equal to, zero, and
then you have plus 2500. So, if you wanted to
maximize this expression, which is the same thing as
maximizing this expression, because they're equivalent, you want, this thing, you wanna
not subtract anything. So, you want this part
to be equal to zero. So, how can this part be equal to zero. Well, the thing that you're
squaring is equal to zero and, in order for the
thing you're squaring to be equal to zero, w
minus 50 needs to be equal to zero, or w needs to be equal to 50. So, just looking at this expression, we're able to figure out 'cause all of these expressions are equivalent and they're equivalent to this
expression right over here. Just looking at this expression we're able to figure out, look, this expression takes on a maximum value when w is equal to 50. And, to their question
which of the following equivalent expression displays, as a constant or a coefficient, the value of the width for
which the area is maximum? We know that the value of the width for which the area's a maximum is 50 and this one I guess displays it. I've never seen a question
phrased exactly like that. But, it does display the number and this is why this form is so valuable. You wanna say, look, this
is going, the area is going to have a maximum value when
this part right over here is not taking anything away from the 2500. In fact, 2500 is going
to be that maximum area. And, when do we reach it? When this thing is zero,
this thing is zero, when w minus 50's equal to zero, or when w is equal to 50. So, I would definitely pick this one here. These other forms of these expressions, and you can algebraically expand these, it'll come out to be the
same thing as over here, these don't help you so much. You can't make that same type of intuition about where the maximum value is. And this form of a quadratic,
this is called vertex form. It makes it very easy to find the vertex, which is going to be the
minimum or the maximum value of an expression, or of
a function I should say. Now, there's other ways
to come up with the 50. You can say, look, this
thing right over here, you have a, it's a quadratic, it's got a negative coefficient
on the quadratic term, or on the second degree
term, so it's going to be a downward opening parabola. So, it might look something like this where if this is the
a-axis, the area axis, and this right over
here is the width axis. That's the width axis. You could, you would know that this is gonna be downward opening because it has this
negative right over here. It's gonna be a downward opening parabola. So, it's gonna look like this. And so, we just need to find the vertex, or we need to find the w coordinate of the vertex there, and
there's a couple of ways that you can find the w
coordinate of the vertex. One, you can say that
that coordinate is going to be half way in between the w values that make our function equal to zero. And so, we can say, well, what makes this thing equal to zero. You could say 100 w minus
w squared is equal to zero. You could factor out a w, so w times 100 minus w is equal to zero. Well, this is going to be equal to zero when either w is equal to zero or 100 minus w's equal to zero. So, or w is equal to, w's equal to 100. If w's equal to 100, this part over here is going to be equal to zero. The whole area is going
to be equal to zero, so this right over here is 100. This one over here is zero. The maximum point then, the vertex, is going to be the w that's half way in between these two,
which is going to be 50. So, that's another way to think about it. You could also use the formula for the x-coordinate of a
vertex, negative b over 2 a. If you had an expression in the form of a w squared plus b w plus c. Then, and then we could say a is equal to, this is a lowercase here, then
the vertex is going to be, the w-coordinate I
should say of the vertex, is gonna be negative b over two a. And, in this case, in this case, b is 100, it's the coefficient
on the first degree term. So, it's going to, if b is 100,
negative b is negative 100. And, a is this negative one that's in front of the w squared. It's the coefficient on
the second degree term. This, they wrote it the other way around where they have the first degree term and then the second degree. Here, you have the second
degree, then the first degree. But, a is negative one. Two times a is negative two. Then, negative divided by
a negative is a positive. 100 divided by two is 50,
so any way you look at it, w equals 50 is when it
hits the maximum value. And, if you looked at these expressions, which of these involve 50 as
a constant or a coefficient? Well, I see a 50 right there. It's a little bit of a
strange way of asking it, but there is a 50 there. It's being subtracted,
but it does show up. None of these other ones have
a 50 anywhere to be seen.