The equation (4 − k) x² + (2k + 4) x + (8k + 1) = 0 has equal roots. Find the values of k.

Given: (4 − k)x² + (2k + 4)x + (8k + 1) = 0

From the given equation, a = (4 − k),  b = (2k + 4),  c = (8k + 1)

An equation has equal roots only if the determinant of the equation is zero.

∴ b² − 4ac = 0

⇒ (2k + 4)² − 4(4 − k) (8k + 1) = 0

⇒ 4 [(k + 2)² − (4 − k) (8k + 1)] = 0

⇒ [k² + 4k + 4 − (31k − 8k² + 4)] = 0

⇒ 9k² − 27k = 0

⇒ k = 0, k = 3

Therefore, option C is the correct answer.