Main content

## SAT (Fall 2023)

### Course: SAT (Fall 2023) > 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

# Solving quadratic equations — Harder example

Watch Sal work through a harder Solving quadratic equations problem.

## Want to join the conversation?

- Is there an easier way of remembering the quadratic formula, because I have a PSAT coming up and I do not understand half of the stuff on there.(23 votes)
- Or, if songs don't do the trick:

A "negative boy" "couldn't decide" if he should go to "radical" party. Being "square", he decided "not" to go and missed out on (4 awesome chicks). The party was "over" at "2a"m.

Hope this helps too!(42 votes)

- would we be asked to solve these kinds of questions without a calculator ?(21 votes)
- It is best to practice as if you will not have a calculator on these. Besides, it is probably easier to solve many of these without a calculator. As Sal says, we have to do a lot of algebraic manipulation to get the formula to a form that we can solve. Until you have the equation in the right form, the calculator can just get in the way.(9 votes)

- Can you upload a different example on another harder question about a different topic under solving quadratic equations please?(16 votes)
- whats the best way to remember the quadratic equation?(7 votes)
- my algebra teacher used to sing to us the quadratic formula to the tune of "pop goes the weasel". ive never forgotten it since. hope that helps (even though im years late)(17 votes)

- I don't understand this. Can someone simplify this for me? If I don't get this, I'm never getting into college. :,((0 votes)
- At the beginning of the video, why do we need to set the equation equal to zero?Could someone please explain.(4 votes)
- because at first you need to multiply both expression and equal to 5.And then after you could combine like terms; that's the way it works.(4 votes)

- How do you know if you multiply through the parenthesis or if you just use the positive one in front of the parenthesis to distribute? I never ;earned how to do this, I always just inserted the different options into the equation to see what my answer would be....but I don't know how to do that with fractions at all.

I am also not very good with square roots either which is another thing I need help with.(3 votes)- why isn't it -20 when you plug it in the quadratic equation?(5 votes)

- Not to sound rude, but aside from SATs or becoming a mathematician or teacher, is there any time we would use this kind of math in day-to-day life? I appreciate Khan Academy and everything they do, but I'm a little scared of coming off as rude. Once again, not my intention, just want to know.(4 votes)
- also where did the 4 come from(3 votes)
- Exactly. The four came from squaring the b-coefficient under the radical. -2 squared is the same thing as 2 squared, both of which equal 4.(3 votes)

- why didn't he subtract 15 from 5?(4 votes)
- If you do that then you are talking about completing the square formula.(1 vote)

## Video transcript

- [Instructor] What are all the solutions to the equation above? And we have x plus
three times x minus five is equal to five. You wanna be very careful here because you're probably
have some experience with algebra that, hey,
once I factored it out, maybe I could say, okay, maybe this needs to be equal to five, or this
needs to be equal to five. That is not the way that it works. That would not actually logical sense. In order to kind of make
the factoring useful, you have to be able to say, hey, the product of these
two things is equal to zero, because if the product of two things is equal to zero, then you know that either one or the both of them need to be equal to zero. So we've actually have to do a lot of algebraic manipulation
here to get it into that form, but let's see if we can do it. So the first thing I
would do is just multiply out x plus three times x minus five. So what's that going to be? Well it's going to be x times x, which is x squared, plus
x times negative five. So I could say that's minus five x. Plus three times x. So that's plus three x. Plus three times negative five. So that's minus 15. And that's all going to be equal to, and that's all going to be equal to five. And let's see what we can do from here. We could say this is gonna be x squared and then these two terms, right over here, we can add together. Negative five x plus
three x is negative two x. And then we have minus
15 is equal to five, is equal to five. Now let's see. We could now subtract
five from both sides. So you subtract five there. You subtract five there. And we're starting to
get to the home stretch. And we would get x squared
minus two x minus 20 is equal to zero. So now we're in business. Now we have this quadratic in a form where we just need to figure out what x values are gonna make
this expression equal to zero. And the first temptation is see well can we, is there, is there any way that we can naturally factor this? So is there a, let's see. Are there any two factors of 20 that if I, and it's going to be, one of them's gonna be positive and one's going to be negative since their product gets us negative, where I add 'em together
I get to negative two. So let's see. Nah, nothing jumps out. You have four and five, two and ten. Yeah, nothin's jumpin' out at me. So we can just resort to
the quadratic formula here. When the quadratic formula tells us that if I have ax squared plus bx plus c is equal to zero, then the solutions of this quadratic equation are going to be x is equal to negative b plus or minus the square root
of b squared minus four ac all of that over two a. And I don't tell people
to memorize a lot in life, but the quadratic formula
is one of those things that it's not a bad idea to memorize. But you should also watch
the Khan Academy videos on how this proves and
this comes just naturally out of completing the square on this thing right over here. So also understand what
it's actually saying but it's also a good thing to know, because just like that
we can apply it to this. So in this context, our a is one. It's the coefficient out here
that's implicitly out there. So a is one. b is negative two. b is equal to negative two. That's this coefficient right over there. And c is negative 20. c is equal to negative 20. So the roots are going to
be x is equal to negative b. So it's gonna be negative of negative two. So negative of negative two
is gonna be positive two, plus or minus the square
root of b squared, which is four, minus four times a, which is one, times negative 20. So instead I'll put a 20 here. And since that's a negative
20 but I'm subtracting it, I could put a plus there. And then all of that over two a. Well a is just one. All of that over two. So it's gonna be two plus
or minus the square root of four plus four times 20 is 80, of 84 over two. Now already, if I'm under time pressure, I already see some
choices that are starting to look a little bit like this. But let's see if we can get
to the right solution here. So can I simplify this more? Well see the square root of 84. 84, that's going to be divisible by 4. 84 is the same thing as four times 21, and all of that over two. And this is going to be the same thing as, I'll scroll down a little bit, get a little bit more space. This is going to, x is going to be equal to two plus or minus. So this would be the same
thing as the square root of four times the square root of 21, which of course is two
times the square root of 21, all of that over two. So if you divide each of these by two, which we are doing right here, it's going to be one plus
or minus the square root of 21, which is this
choice right over there. So it looked like a fairly benign thing, but we had to multiply it out, set it up in kind of a form
where the quadratic formula would apply, and we got
a fairly hairy answer.