The standard form for the equation of a circle is:
#(x - h)^2 + (y - k)^2 = r^2#
Substitute #(y - 0)^2# for #y^2#
#x^2 + 14x + (y - 0)^2 = 0#
Using the pattern #(x - h)^2 = x^2 - 2hx + h^2#, we observe that we must begin the process of completing the square by adding #h^2# to both sides:
#x^2 + 14x +h^2 + (y - 0)^2 = h^2#
For this circle, the #h^2# term is, also the #r^2# term, because the we will soon find that the center is offset to the left the same distance as the radius.
We can find the value #h# by setting the right side of the pattern equal to the first 3 terms in the equation:
#x^2 - 2hx + h^2 = x^2 + 14x +h^2#
The square terms cancel:
#-2hx = 14x#
#h = -7 and h^2 = 49
Substitute #(x - -7)^2# for the corresponding terms on the left and
#(x - -7)^2 + (y - 0)^2 = 49#
The radius should be represented as a positive number squared.
#(x - -7)^2 + (y - 0)^2 = 7^2#
The center is found by observation #(-7, 2)#
