It will touch the x-axis when y=0 so:
3x^2-5x+1=0 subtract 1 from both sides
3x^2-5x=-1 divide all terms by 3
x^2-5x/3=-1/3 halve the linear coefficient, square it, then add that to both sides....(5/6)^2=25/36
x^2-5x/3+25/36=25/36-12/36 now the left side is a perfect square
(x-5/6)^2=13/36
(x-5/6)^2=1/3 take the square root of both sides
x-5/6=±√(1/3) add 5/6 to both sides
x=5/6±√(1/3)
So the graph will intersect the x-axis at two points when:
x=5/6-√(1/3) and 5/6+√(1/3)
