Find the range of values of K for which the equation X^2+kx+x-K^2+1=0 has real roots?

x^2 + kx + x - k^2 + 1 = 0
x^2 + (k + 1) * x - (k^2 - 1) = 0

a = 1
b = k + 1
c = -(k^2 - 1)

b^2 - 4ac >/= 0
(k + 1)^2 + 4 * 1 * (k^2 - 1) >/= 0
(k + 1) * (k + 1 + 4 * (k - 1)) >/= 0
(k + 1) * (k + 1 + 4k - 4) >/= 0
(k + 1) * (5k - 3) >/= 0
5 * (k + 1) * (k - 0.6) >/= 0

-1 < k < 0.6

-inf < k </= -1
0.6 </= k < inf

Those are the values

x² + (k+1)x + (1-k²) = 0
two real solutions when discriminant > 0

discriminant = (k+1)² - 4·1·(1-k²)
= (k² + 2k + 1) - (4-4k²)
= k² + 2k + 1 - 4 + 4k²
= 5k² + 2k - 3
discriminant > 0 when -1 < k < 0.6

k = -1 <x< 5/3