Determine the points that are on the curves y=x^2+x+1 and y=-x^2-2x+6.

To determine the points that lie on both the curves
y=x^2+x+1 and y=-x^2-2x+6, we have to equate the two. Doing this
gives

x^2 + x + 1 = -x^2 - 2x +
6

=> x^2 + x^2 + x + 2x + 1 - 6 =
0

=> 2x^2 + 3x - 5 =
0

=> 2x^2 + 5x - 2x - 5
=0

=> x ( 2x + 5) - 1( 2x + 5) =
0

=> ( x -1)( 2x + 5) =
0

This gives x = 1 and x =
-5/2

At x = 1, y = x^2 + x + 1 = 1 + 1 + 1 =
3

At x = -5/2 , y = (-5/2)^2 - 5/2 + 1 =
19/4

Therefore the points that lie on both
the curves are ( 1, 3) and ( -5/2 ,
19/4).