The sum of all integral values of k ( k ≠ 0 ) for which the equation 2 x - 1 - 1 x - 2 = 2 k in x has no real roots, is_____.

Step-by-step solution

We have, 2 x - 1 - 1 x - 2 = 2 k ⇒ 2 x - 4 - x + 1 ( x - 1 ) ( x - 2 ) = 2 k ⇒ x - 3 x 2 - 3 x + 2 = 2 k ⇒ k x - 3 k = 2 x 2 - 6 x + 4 ⇒ 2 x 2 - ( 6 + k ) x + 3 k + 4 = 0 For no real roots D < 0 ⇒ ( 6 + k ) 2 - 8 ( 3 k + 4 ) < 0 ⇒ k 2 + 12 k + 36 - 24 k - 32 < 0 ⇒ ( k - 6 ) 2 - 32 < 0 ⇒ k - 6 < 32 ⇒ 6 - 32 < k < 6 + 32 Integral value of k = 1 , 2 , 3 , 4 ,   5 , 6 , 7 , 8 , 9 , 10 , 11 Sum = 11 × 12 2 = 66