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

x² - 2(5 + 2k)x + 3(7 + 10k) = 0

Comparing x² - 2(5 + 2k)x + 3(7 + 10k) = 0 with ax² + bx + c = 0 we get,

a = 1, b = -2(5 + 2k) and c = 3(7 + 10k).

Since equations has equal roots,

∴ D = 0

⇒ [-2(5 + 2k)]² - 4 × 1 × 3(7 + 10k) = 0

⇒ 4(5 + 2k)² - 12(7 + 10k) = 0

⇒ 4[(5)² + (2k)² + 2 × 5 × 2k] - (84 + 120k) = 0

⇒ 4(25 + 4k² + 20k) - 84 - 120k = 0

⇒ 100 + 16k² + 80k - 84 - 120k = 0

⇒ 16k² - 40k + 16 = 0

⇒ 16k² - 8k - 32k + 16 = 0

⇒ 8k(2k - 1) - 16(2k - 1) = 0

⇒ (2k - 1)(8k - 16)= 0

⇒ (2k - 1) = 0 or (8k - 16)= 0      [Using Zero-product rule]

⇒ 2k = 1 or 8k = 16

⇒ k = $\dfrac{1}{2}$ or k = $\dfrac{16}{8}$

⇒ k = $\dfrac{1}{2}$ or k = 2

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