Question 1187043
First best to find the intersection points of the two quadratic equations.


{{{system(3y=-x^2-2x-4,y=-x^2+3x-3)}}}


{{{system(3y=-x^2-2x-4,3y=-3x^2+9x-9)}}}


Equate the expressions for 3y.
{{{system(-x^2-2x-4=-3x^2+9x-9)}}}


{{{2x^2-11x+5=0}}}


discrim, {{{121-8*5=121-40=81=9^2}}}

zeros of the quadratic eq.
{{{system((11-9)/4,and,(11+9)/4)}}}

{{{system(1/2,and,5)}}}----------------(could also have easily been factored if really wanted)


Find the corresponding y values.
{{{y=-3x^2+3x-3}}}
IF x at {{{1/2}}}, {{{y=-3/4+3/2-3=-3/4+6/4-12/4=-9/4}}}
IF x at 5, {{{y=-3*25+15-3=-75+15-3=-60-3=-63}}}


The two given parabolas intersect at  (1/2, -9/4)  and (5, -63).


Now, you want to find equation for the parabola with vertical symmetry axis and containing the three points  (0,2), (1/2, -9/4), and (5, -63).   You could setup three equations using these points starting in a format  {{{ax^2+bx+c=y}}}, and solve the system.   You continue with that.