Arocket is launched from the ground and follows a parabolic path represented by the equation y= -x squared + 10x. at the same time, a flare is launched from the height of 10 feet and follows a straight path represented by the equation y= -x + 10. will these two object interest? if they do interest, where will it be?

X=horizontal position
y=vertical position (height)
y=-x²+10x
y=-x+10
intersect is when the position (x,y) on the path is the same on both

set equal to each other
-x²+10x=-x+10
times -1 both sides
x²-10x=x-10
add 10 to both sides and minus x both sides
x²-9x+10=0
factor
(x-10)(x+1)=0

set each to zero
x-10=0
x=10

x+1=0
x=-1
false, no negative positions (assuming it didn't go backwards)

x=10
sub into a function
y=-x+10
y=-10+10
y=0

yes they intersect at (10,0), or at 10 feet to the right and 0 feet up