Solve the following quadratic by completing the square; x^2+11x+1=0

x^2+11x+1=0  subtract 1 from both sides

x^2 + 11x  =  - 1        take (1/2) the coefficient on x, (11), square it and add to both sides..so (11/2) = 121/4 and we have

x^2 + 11x + 121/4 = -1 + 121/4      factor the left side and simplify the right

(x + 11/2)^2 = 117/4        take the ± square root of both sides

x + 11/2  = ±(√117)/2      subtract 11/2 from bot sides and note that √117 = √(13*9) =3√13....so we have..

x = [ - 11 ± 3√13] / 2