K has a range from[ -1/11 to 1]

real roots occur when the determinant b^2-4ac >= 0 so substituting:- (5k + 1)^2 - 4*1* (9k^2) >= 0 25k^2 + 10k + 1 - 36k^2 >= 0 -11k^2 +10k + 1 >= 0 11k^2 - 10k - 1 <= 0 (11k + 1)(k - 1) <= 0 the equation will have real roots for -1/11 =< k <= 1