Hi! Basically, the problem is asking us to find the values of a, b, and c in the equation
![f(x)=a x^{2} +bx+c](https://tex.z-dn.net/?f=f%28x%29%3Da%20x%5E%7B2%7D%20%2Bbx%2Bc)
. Since we have three unknowns, we just need three equations. We can find these equations by using the data in the table.
First let's plug x = 0 and f(x) = 0.
![0=0+0+c](https://tex.z-dn.net/?f=0%3D0%2B0%2Bc)
![c=0](https://tex.z-dn.net/?f=c%3D0)
Now that we know c, it's time to pick two more pairs. Let's plug-in (2,78) and (4,152)
![78=4a+2b](https://tex.z-dn.net/?f=78%3D4a%2B2b)
![152=16a+4b](https://tex.z-dn.net/?f=152%3D16a%2B4b)
Before proceeding with the process of eliminating one variable, let us first reduce both equations to their lowest terms. We divide the first equation by 2 and we divide the second one by 4.
![39=2a+b](https://tex.z-dn.net/?f=39%3D2a%2Bb)
![38=4a+b](https://tex.z-dn.net/?f=38%3D4a%2Bb)
Next, we subtract equation 2 from equation 1.
![1=-2a](https://tex.z-dn.net/?f=1%3D-2a)
![a=-0.5](https://tex.z-dn.net/?f=a%3D-0.5)
Finally, we substitute the value of a to equation 2 to get the value of b.
![38=4(-0.5)+b](https://tex.z-dn.net/?f=38%3D4%28-0.5%29%2Bb)
![b=40](https://tex.z-dn.net/?f=b%3D40)
Therefore, the function should be