Writing a quadratic function from solutions
About this video
This video finds standard-form equations for a quadratic with solutions x = -3 and x = 5. It writes f(x) = (x + 3)(x - 5), expands to x^2 - 2x - 15, and shows that multiplying by any non-zero number, such as 2 or -2, gives different functions with the same zeros.
You'll be able to
- write a quadratic as (x - first solution)(x - second solution).
- expand that product into standard form.
- explain why multiplying a quadratic by a non-zero constant keeps the same zeros but changes the function.
Key words
- Solution (zero)
- An x-value that makes f(x) equal zero.
- Standard form
- A quadratic written as ax^2 + bx + c.
- Coefficient
- A number multiplying a term, like the 2 in 2x^2.
- Quadratic function
- A function whose highest power of x is x^2.
Transcript
- [Tutor] We're told A quadratic function F has two real solutions, X equals negative three and X equals five that make F of X equals zero. 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.
Alright, 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 minus the first solution. And the first solution here is when X is equal to negative three and then times X minus the second solution when X is equal to five.
Now why does this work? Well, think about it if X is equal to negative three right here and if I were to subtract another negative three, well then this is going to be equal to zero. Zero times anything is zero and then F of negative three would be zero. Similarly, if X were equal to five here, well then this whole thing would be equal to zero.
Zero times anything is zero. So, F of five 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 squared, plus some other constant times BX plus some other constant. So, to get there we have to multiply this out.
And actually before we do that, lemme just simplify it a little bit. This is going to be equal to X. When I subtract a negative three, that's the same thing as adding three and then times X minus five. So, X plus three times X minus five. 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 squared. We have X times negative five, which is negative 5x. We have three times X, which is 3x, and then we have three times negative five, which is minus 15 or negative 15, or we're gonna subtract 15 here.
And so last but not least, we have X squared. And if I'm subtracting 5x and then I add 3x, that is negative 2x minus 15. So, this is F of 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 two or negative two.
I don't see that over here. So, what is happening here is I can multiply this whole thing by a two or a negative two and it's not going to change where my zeros are. Why is that? Well, think about it. If I had a two over here, when X is equal to five, this is gonna be zero times something times two, it's still going to be equal to zero.
Similarly, if that were a negative two, so I'm gonna have the same zeros if I multiply it by really any number that is not zero. So, let's do that. So, if I were to multiply this equation by positive two, I need to multiply all of them by two. I'm running outta space.
So, I'll do it up here. We would get F of X is equal to two times X squared is 2x squared. Two times negative 2x is negative 4x. Two times negative 15 is negative 30. That's one way we could think about it. Another way we could say maybe F of X is going to be equal to.
And to be clear, these are not the same functions when I multiply it by two or negative two. It does fundamentally change the function, but they would have the same zeros, it would have the same two real solutions that X equals negative three and X equals five.
So, if I were to say, well, maybe instead of this F of X, could be this times negative two. Once again, it's a different F of, it's a different function in these situations. I'm just trying to find out all of the possibilities and there could be many more. I could multiply it by three or negative three or anything else.
But if I were to multiply this by negative two, I would get negative 2x squared, negative two times negative 2x is positive 4x. Negative two times negative 15 is positive 30. So, I'm gonna 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 equals negative three and X equals five. So, now let's see which choices match up. 2x squared minus 4x minus 3O. 2x squared minus 4x minus 30. I like this one right here.
And the negative 2x squared plus 4x plus 30, negative 2x squared plus 4x plus 30. I like this one here as well. So, I'm done.
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