Find the coordinates of the intersection

Question

Find the coordinates of the intersections of

y1=x-(x^3/6)+(x^5/120)

y2=x^2


Make a sketch of the functions on the grid below. Estimate the x,y coordinates to decimal places. There are 3 intersections. The third intersection will be out of range of the grid provided, so just show the first 2 intersections but be sure to state all three intersections in your final answer.

Answer 1

y= x - (x^3)/6 + (x^5)/120

y = x^2

intersection points are when

(x^5)/120 - (x^3)/6 - x^2 +x = 0

one such a point is (0,0) by inspection, then

x^4)/120 - (x^2)/6 -x +1 = 0

x^4 - 20x^2 -120x +120 = 0

by Descartes sign rule there's 0 or 2 + real solutions= 0 or 2 Positive x intersection points

and 0 or 2 - real solutions = 0 or 2 negative x intersection points, and 0,2 or 4 imaginary solutions

solve by graphing the 4 degree equation in x

2 + x solutions, 0 - x solutions, 2 imaginary solutions

x= about .88 and 6.05, y = .77, 36.60

the 3 intersection points are

(0,0), about (.88, .77) and (6.05, 36.60)