The method of completing the square is based on the fact that there is a pattern in the answer of squaring a binomial.
#(x - 5)^2 = x^2 - 10x +25 " Note "(10/2)^2 = +25#
The square of (half of the coefficient of x), gives the last term.
So let's find the missing term ....
#x^2 - 14x + ......color(red)(??).... rArr x^2 - 14x + color(red)(7^2)#
#x^2 + 22x + .....color(blue)(??).. rArr x^2 + 22x + color(blue)(11^2)#
Once you have all three terms, you can write them as #(x +- ...)^2#
#x^2 - 14x + color(red)49 = (x -color(red)7)^2 " -14÷2 = -7, or "sqrt49 =7#
#x^2 + 22x + color(blue)121 = (x + color(blue)11)^2#
Now for your question....
#x^2 -8x +13 =0 " move 13 to the right side"#
#x^2 -8x + 16 =-13 +16 " add 16 to both sides"#
#(x-4)^2 =3 " Write as a square"#
#x - 4 = +-sqrt3" square root both sides"#
#x = sqrt3 +4 or x = -sqrt3+4" solve for x twice"#
#x = 5.732 or x = 2.268#
