Does the equation x2 - 10x + y2 - 6y = - 30 intersect the x-axis?

Intersecting the x axis means y is 0

plug y=0 in the equation and solve for x to see if there are real solutions

x^2-10x+0+0=-30

x^2-10x+30=0

this cannot be factored, so let's see if b^2-4ac is bigger than 0

10^2-4*1*30=-20

so the solutions for this equation are not real numbers, therefore the answer is no.