Harristotelian Logic

Consider

x^2 - 5x + 2 = 0

and

y^2 - 7y + 2 = 0

and solve each where I'll not use +/- and just use plusses:

x = (5 + sqrt(21))/2 and y = (7 + sqrt(45))/2.

I can subtract

x^2 - 5x + 2 = 0

from

y^2 - 7y + 2 = 0

to get

y^2 - x^2 = 7y - 5x

and group a bit, and divide by y - x to get

y + x = 2y/(y-x) + 5

so

(7 + sqrt(45))/2 + (5 + sqrt(21))/2 = 2(5 + sqrt(21))/(2 + sqrt(45) - sqrt(21)) + 5

which proves that 2 + sqrt(45) - sqrt(21) must be a factor of 2.

Just doodling.

Just curious, what is the polynomial with integer coefficients that has that as a root?