We must complete the square to put the equation into vertex form: y=a(x−h)2+k, where (h,k) is the vertex.
y=−7(x2+27x+?)+3
We must complete the square. In order to do this, we must recall that (x+a)2=x2+2ax+a2, so the middle term, 27x, is 2x times some other number, which we can determine to be 17. Thus, the final term must be (17)2.
y=−7(x2+27x+149)+3+17
Note that we had to balance the equation—we can add numbers randomly. When the 149 was added, we must realize that it is actually being multiplied by −7 on the outside of the parentheses, so it is actually like adding −17 to the right side of the equation. In order to balance the equation we add a positive 17 to the same side.
Now, we can simplify:
y=−7(x+17)2+227
Since the vertex is (h,k), we can determine its location is (−17,227). (Don't forget the h value switches signs.)
