The equation which solves the graph of system of equation is:
[tex]x^2+6x+8=-x^2-8x-16[/tex]
(-4,0),(-3,-1) and (-2,0)
Let the equation of parabola be:
[tex]y=ax^2+bx+c[/tex]
Now, when we take the point (-4,0) we have:
[tex]16a-4b+c=0----------(1)[/tex]
when we take the point (-3,-1) we have:
[tex]9a-3b+c=-1---------(2)[/tex]
and when we take the point (-2,0) we have:
[tex]4a-2b+c=0-----------(3)[/tex]
on subtracting equation (3) from equation (1) we have:
[tex]12a-2b=0\\\\i.e.\\\\12a=2b\\\\i.e.\\\\b=\dfrac{12a}{2}\\\\i.e.\\\\b=6a[/tex]
and on putting the value of b in equation (3) we have:
[tex]4a-2(6a)+c=0\\\\i.e.\\\\4a-12a+c=0\\\\i.e.\\\\c=8a[/tex]
Now, on putting the value of b and c in terms of a in equation (2) we have:
[tex]9a-3(6a)+8a=-1\\\\i.e.\\\\9a-18a+8a=-1\\\\i.e.\\\\-a=-1\\\\i.e.\\\\a=1[/tex]
Hence,
[tex]b=6\\\\and\\\\c=8[/tex]
Hence, the equation of blue parabola is:
[tex]y=x^2+6x+8[/tex]
(-4,0) , (-3,-1) and (-5,-1)
Hence, the three equations are:
[tex]16a-4b+c=0----------(1)\\\\9a-3b+c=-1----------(2)\\\\25a-5b+c=-1-----------(3)[/tex]
on solving the three equations we have:
[tex]a=-1\\\\b=-8\\\\and\\\\c=-16[/tex]
Hence, we have the equation of the red parabola as:
[tex]y=-x^2-8x-16[/tex]
Hence, the equation of the graph that need to be solved is:
[tex]x^2+6x+8=-x^2-8x-16[/tex]
