Question

Find the value of k for which the following equation has equal roots:
(k − 12)x² + 2(k − 12)x + 2 = 0.

Answer 1

The given equation is

(k − 12)x² + 2(k − 12)x + 2 = 0

Here, a = k − 12, b = 2(k − 12) and c = 2

Since the given equation has two equal real roots

then we must have b² − 4ac = 0

⇒ [2(k − 12)]² − 4(k − 12) x 2 = 0

⇒ 4(k − 12)² − 8(k − 12) = 0

⇒ (k − 12) (k − 12) − 2 (k − 12) = 0   ...(dividing by 4)

⇒ (k − 12)[(k − 12) − 2] = 0

⇒ k − 12 = 0 or k − 14 = 0

⇒ k = 12 or k = 14.
Note: But at k = 12, terms of x² and x in the equation vanish; hence, only k = 14 is acceptable.