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
The Housing Market Is Going Insane | $0 DOWN MORTGAGES ARE BACK
What’s up, Graham? It’s guys here. So, with housing prices beginning to decline, mortgage rates approaching 6%, and home buyer affordability falling to its worst level in 37 years, banks have started to once again bring back no money down loans for certai…
Quiet Quitting Is Going To Ruin Your Career | Shepard Smith
You’re introducing a cancer into your culture; eventually, you’re going to have to do surgery and cut it out. I don’t know where this started; it’s the worst idea I’ve ever heard. [Applause] [Music] So, quiet quitting: a temporary pandemic hangover, bypr…
The Real Moral Dilemma of Self-Driving Cars
Push this button. It’s driving itself. It feels good. So, BMW brought me to the Consumer Electronics Show here in Las Vegas. I’m going to check out the future of driving. Did I get it? Am I near? [unintelligible] Oh! I felt it! That really felt like pushi…
How Will The Federal Reserve Stop Inflation?
[Music] At the most recent meeting of the Federal Reserve Open Market Committee, it was forecast that inflation is due to rise, and they signaled that as a result, rate increases might move forward sooner than they expected. Now, I explained all this in a…
Unlocking the Eyes | Explorer
[Music] What boggles my mind about the eye is everything. But I’m really, really excited by the advances in technology made possible by research, not just into the eye, but into how natural selection caused it to be what it is. The next few decades are go…
The reason I built the worlds first private jet showroom!
The reason I built the first and only Aviation showroom in the world is because nobody else has. I had to be different. Everybody in our industry today lives off a mobile phone and a laptop; that’s a business, that’s their office. To me, it just doesn’t s…