What is the equation of the quadratic function represented by this table?

Step-by-step explanation:

I used logic and took the easy way around this as opposed to the long, drawn-out algebraic way.  I noticed right off that at x = -3 and x = -1 the y values were the same.  In the middle of those two x-values is -2, which is the vertex of the parabola with coordinates (-2, 4).  That's the h and k in the formula I'm going to use.  Then I picked a point from the table to use as my x and y in the formula I'm going to use.  I chose (0, 3) because it's easy.  The formula for a quadratic is

and I have everything I need to solve for a.  Filling in my h, k, x, and y:

 and

 and

-1 = 4a so

In work/vertex form the equation for the quadratic is

In standard form it's:

16

Step-by-step explanation:

We must find integers x, y with the most amount of prime divisors, not necessarily distinct, such that x + 3y < 1,000.

Obviously, this is achieved when the divisor is the least prime 2. So, we must find integers n, m such that

since , then n must be 9. For n=9 we find the greatest integer m such that  

and we find m=7

 and   ,  are the numbers we are looking for and the sum of their length is 9+7 = 16.

So, 16 is the maximum possible sum of the length of x and the length of y.