yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Writing a quadratic function from solutions | Algebra 1 (TX TEKS) | Khan Academy


3m read
·Nov 10, 2024

We're told a quadratic function ( f ) has two real solutions ( x = -3 ) and ( x = 5 ) that make ( f(x) = 0 ). Select the equations that could define ( f ) in standard form. So, pause this video and have a go at that before we do this together.

All right, so there's a bunch of ways you could approach this, but the way that I think about it is we can express this quadratic in terms of its two solutions. So, you could have ( x - ) the first solution, and the first solution here is when ( x ) is equal to -3, and then times ( x ) us the second solution when ( x ) is equal to 5.

Now, why does this work? Well, think about it. If ( x ) is equal to -3 right here, and if I were to subtract another -3, well then this is going to be equal to 0. ( 0 \times ) anything is zero, and then ( f(-3) ) would be zero. Similarly, if ( x ) were equal to five here, well then this whole thing would be equal to zero; ( 0 \times ) anything is 0, so ( f(5) ) is zero.

Now, this is a definition of the quadratic, but it is not in standard form. Standard form, as a reminder, would be some constant times ( x^2 ) plus some other constant times ( x ) plus some other constant. So, to get there, we have to multiply this out.

And actually, before we do that, let me just simplify a little bit. This is going to be equal to ( x ) when I subtract a -3. That's the same thing as adding three, and then times ( x - 5 ). So, ( x + 3 \times x - 5 ).

And now we can expand this out so we get it to standard form. So, this is going to be equal to ( x \times x ), which is ( x^2 ). We have ( x \times -5 ), which is -5x. We have 3 times ( x ), which is 3x, and then we have 3 times -5, which is -15.

So, last but not least, we have ( x^2 ), and if I am subtracting 5x and then I add 3x, that is -2x minus 15. So, this is ( f(x) ) in standard form.

Now, let's see which of these choices gets me this. So when I look over here, well, what's interesting is all of these have a coefficient of either 2 or -2. I don't see that over here. So what is happening here is I can multiply this whole thing by 2 or -2, and it's not going to change where my zeros are.

Why is that? Well, think about it. If I had a 2 over here, when ( x ) is equal to 5, this is going to be ( 0 \times ) something (\times 2); it's still going to be equal to zero. Similarly, if that were a negative -2, so I'm going to have the same zeros if I multiply it by really any number that is not zero.

So let's do that. If I were to multiply this equation by positive 2, I need to multiply all of them by two. I'm running out of space, so I'll do it up here. We would get ( f(x) ) is equal to ( 2 \times x^2 ), which is ( 2x^2 ); ( 2 \times -2x ) is -4x; ( 2 \times -15 ) is -30. That's one way we could think about it.

Another way we could say maybe ( f(x) ) is going to be equal to, and to be clear, these are not the same functions. When I multiply it by 2 or -2, it does fundamentally change the function, but they would have the same zeros; they would have the same two real solutions ( x = -3 ) and ( x = 5 ).

So if I were to say, “Well, maybe instead of this, ( f(x) ) could be this times -2,” once again, it's a different ( f(x) ); it's a different function. In these situations, I'm just trying to find out all the possibilities, and there could be many more. I could multiply it by 3 or -3 or anything else.

But if I were to multiply this by -2, I would get ( -2x^2 ); ( -2 \times -2x ) is ( +4x ); ( -2 \times -15 ) is ( +30 ). So I’m going to say it one more time, the three things that I’m boxing off here, these three possible functions, these are all different functions. If I were to graph it, they would all look different, but they all have the same two real solutions ( x = -3 ) and ( x = 5 ).

So now, let's see which choices match up: ( 2x^2 - 4x - 30 ), ( 2x^2 - 4x - 30 ). I like this one right here and then ( -2x^2 + 4x + 30 ), ( -2x^2 + 4x + 30 ). I like this one here as well, so I'm done.

More Articles

View All
Steve Varsano talks about his experience in aviation
When you’re selling a jet for a company, that company is either moving up to a bigger, newer jet, or the company’s having problems and they’re selling the jet and they’re getting out of the business of operating their own corporate jet. If it’s the latte…
Why “Looking Poor” Is Important
What’s up you guys, it’s Graham here. In the last few months, you might have come across one of these videos: the importance of looking poor. After all, when you really dig into it, it is insane how many people these days are pretending to be rich, diggi…
I Need Your Help!
That echo, that is a nasty echo. Anyway, um, hello! Welcome to New Money HQ. This is pretty exciting, isn’t it? Um, so as you can see, I am currently in quite an empty office space and, uh, well, this is one of the parts of, uh, the expansion of the chann…
last words
Hey, Vsauce. Michael here. On December 17th, 1977, Gary Gilmore was executed for murder. He was the first prisoner executed by the United States after a 10-year suspension of the practice. When asked if he had any last words, he simply replied, “let’s do…
RECESSION ALERT: The FED Just Crashed The Stock Market
Welp, I thought this is going to be a normal day. As I woke up, opened my computer, took a sip of coffee, expected to get more recommendations on Johnny Depp’s trial, and was immediately hit by the headline: GDP fell by 1.4 percent, leading to the concern…
Finding Something to Live and Die For | The Philosophy of Viktor Frankl
“The meaning of life is to give life meaning.” What keeps a human being going? The purest answer to this question is perhaps to be found in the worst of places. Austrian psychiatrist, philosopher, and author Viktor Frankl spent three years in four differe…