A rocket 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

By doing the math, all I know is at they will intersect but I am not sure where.