For what values of k does the equation x^2 - (5k+1) x + 9k^2 = 0 have real roots?

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

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