ElectricPavlov

horizontal line is   y=8

vertical line is x =4

they intersect at      4,8

put '6' in for height in the equation given    ( which is a dome shaped parablola due to the negative coeeficient of t²)

6 = -4.9 t^2 + 14t - .4

0 = - 4.9t^2 + 14t  - 6.4          now use Quadratic Formula to calculate the two 't values 

               ....the ball is above 6  between thes two values , so calculate the time between these two values of t

a = -4.9     b = 14      c = -6.4

\(t = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

Where they intersect , they are equal....

2x^2 - 7x + 1        =        8x^2 + 5x + 1       gather all terms to one side and simplify

6x^2 + 12x  = 0

6x ( x+2)  = 0        so the   'x' coordinates of the two points of intersection are   0   and  -2

                              use these values of 'x' in either of the equations to calculate the corresponding 'y' coordinates of the intersection points

The green line in the image below represents f(c) = -1

    everywhere the line crosses the red graph is a value of c   that is   f(c) = -1     How many times does the line cross the red graph?

Lookat x = -2       now go vertical to intercept the graph  at  y= ~ 2/3      this is   f(-2)

This is the equation of a circle ......re write in standard form by 'completing the square' for x and y :

2 x^2 + 10x       + 2y^2 -6y  - 18 = 0      divide through by 2

x^2 + 5x            + y^2 -3y      = 9          complete the squares

(x+2.5)^2              + ( y-1.5)^2     = 9 + 6.25  + 2.25   <======== this equals radius ^2

                                                 17.5 = r^2           area of circle = pi r^2        area =   17.5 pi  units²

Where they intersect they are equal

-x^2 -x +1       =       2x^2 -1       gather 'like' terms and simplify

3x^2 +x -2 = 0          Use Quadratic Formula to find    x = 2/3  and -1        these are   c   and   a    in your question

Take it from here !  

You are welcome !!

Basically this is a parbola with the zeroes at -2   and 3    .......it is negative in between these two points (less than zero)

(x+2)(x-3) = x^2 -x - 6 

   (you can calc b+c from here)

here is the graph:

https://www.desmos.com/calculator/yofniiyxqn

Resultant x - axis force is

110 cos 55   +   165 cos 42     = x component

y components

  110 sin 55   + 165 sin 42    = y component

Total magnitude will be    sqrt ( y component²   + x component² )                 (Pythagorean theorem)

angle of the force will be    arctan ( y component / x component)