Relative Motion on a Circular Track

Alex, Ben and Charles were running on a 400 m circular track.

Alex started running at point \(O\) in an anti-clockwise direction at a speed of \(x\) m/s.

At the same time, Ben and Charles also started running at point \(O\), but moved in a clockwise

direction at speeds of \(\left( x+3 \right)\) m/s and \(\left( x-4 \right)\)m/s respectively.

1. Show that the time passed before Alex and Ben meet each other on the track is \(\frac{400}{2x+3}\) seconds.
2. Find, in terms of \(x,\) the time passed before Alex and Charles meet each other on the track.
3. Given that Alex meets Charles \(23\frac{1}{3}\) seconds after passing Ben, form an
   equation in terms of \(x\) and show that it simplifies to \(2{{x}^{2}}-x-66=0\).
   
4. Solve the equation \(2{{x}^{2}}-x-66=0\).
5. Find the time taken for Alex to run one round around the track.