This came up in another thread.
We need to find the coordinates of M when BA is a maximum. Geogebra to the rescue!
1)The equation of the parabola can be determined by symmetry. It is:
2) Enter into the input bar f(x) = -2*x^2 - 4*x and press enter.
3) Enter into the input bar g(x) = x+2 and press enter.
4) Create and label 3 points O(0,0), C(-1,2), D(-2,0).
5) Pick a point that is on the left of C and place it on the parabola. Label it B.
6) Now drop a perpendicular line from B to the x axis.
7) Find the intersection point between this perpendicular line from B to the line x+2. Call this point of intersection A. ( use the intersection tool ).
8) Use the distance tool to measure the distance from B to A. It will appear in the algebra pane as distanceBA = something.
9) Set rounding to 10 digits.
10) use the move tool to slide point B along the parabola. See how close you can come to 1.125, I got 1.1249967. That is the maximum length of BA.
11) Go into the algebra pane and find x = -1.24941724. That is the x coordinate you seek.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
No great discovery was ever made without a bold guess.