Hi! Basically, the problem is asking us to find the values of a, b, and c in the equation [tex]f(x)=a x^{2} +bx+c[/tex]. 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.
[tex]0=0+0+c[/tex]
[tex]c=0[/tex]
Now that we know c, it's time to pick two more pairs. Let's plug-in (2,78) and (4,152)
[tex]78=4a+2b[/tex]
[tex]152=16a+4b[/tex]
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.
[tex]39=2a+b[/tex]
[tex]38=4a+b[/tex]
Next, we subtract equation 2 from equation 1.
[tex]1=-2a[/tex]
[tex]a=-0.5[/tex]
Finally, we substitute the value of a to equation 2 to get the value of b.
[tex]38=4(-0.5)+b[/tex]
[tex]b=40[/tex]
Therefore, the function should be [tex]f(x)=-0.5 x^{2} +40x[/tex]
