Find the values of k for which the following equation has equal roots:

x² + 4kx + (k² - k + 2) = 0

Comparing x² + 4kx + (k² - k + 2) = 0 with ax² + bx + c = 0 we get,

a = 1, b = 4k and c = (k² - k + 2).

Since equations has equal roots,

∴ D = 0

⇒ (4k)² - 4.1.(k² - k + 2) = 0

⇒ 16k² - (4k² - 4k + 8) = 0

⇒ 16k² - 4k² + 4k - 8 = 0

⇒ 12k² + 4k - 8 = 0

⇒ 12k² + 12k - 8k - 8 = 0

⇒ 12k(k + 1) - 8(k + 1) = 0

⇒ (k + 1)(12k - 8) = 0

⇒ (k + 1) = 0 or (12k - 8) = 0      [Using Zero-product rule]

⇒ k = -1 or 12k = 8

⇒ k = -1 or k = $\dfrac{8}{12}$

⇒ k = -1 or k = $\dfrac{2}{3}$.

Hence, k = $\Big{-1,\dfrac{2}{3}\Big}$.