click here to see the graph and the intersection of the two graphs:

from 2x-y-10=0 follows y = 2x - 10

plug y = 2x - 10 into y^2-10x+10y-55=0

(2x - 10)^2 - 10x + 10(2x - 10) - 55 = 0

by solving we find:

x1 = 9.02

x2 = -1.52

click here to see the step by step solution of the quadratic equation:

for x1 = 9.02 we have y1 = 2*9.02 - 10 = 8.04

for x2 = -1.52 we have y2 = 2*(-1.52) - 10 = -13.04

the points of intersection are: (9.02, 8.04) and (-1.52, -13.04).