For what value of k, are the roots of the quadratic equation y² + k² = 2 (k + 1) y equal?

Given, y² + k² = 2 (k + 1) y

⇒ y² – 2 (k + 1) y – k² = 0

For roots to be equal, the value of discriminant must be zero i.e.,

D = b² – 4ac = 0

⸫ D = (-2k – 2)² – 4(1) (-k²) = 0

⇒ 4k² + 4k + 4 + 4k² = 0

⇒ 8k² + 4k + 4 = 0

⇒ 2k² + k + 1 = 0

⇒ 2k² + 2k – k + 1 = 0

⇒ 2k (k + 1) – 1 (k + 1) = 0

⇒ (k + 1) (2k – 1) = 0

⇒ k + 1 = 0 and 2k – 1 = 0

⸫ k = -1 and k =