Find coordinates of the intersections

Question

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

 

y2=x^2

Answer 1

(0,0) is an obvious intersection.

Just set Y1=Y2 and solve for x

x^5/120 - x^3/6 + x = x^2

x^5 - 20x^3 -120x^2 + 120x = 0

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

set each factor = 0 and solve for x

x=0 gives one intersection of (0,0), the origin

x^4-20x^2 - 120x + 120 = 0 has potentially 4 real solutions

but if it looks as if there are no real solutions except in the interval 0<x<1

It seems to be slightly more than .9

Y2=x^2 = .9^2 = .81

the intersections are (0,0) and about (.9,.81)