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