Find the interval in which $k$ must lie in order that the roots of $12 (k + 2) x^{2} - 12 (2 k - 1) x - 38 k - 11 = 0$ may be imaginary.

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Solution

$\begin{matrix} 12 (k + 2) x^{2} - 12 (k - 1) x - 38 k - 11 = 0 a = 12 (k + 1) b = - 12 (2 k - 1) c = - 38 k - 11 f o r r o o t s t o b e i m a g i n a r y = b^{2} - 4 a c < 0 \Rightarrow {(- 12 (2 k - 1))}^{2} - 4 \times 12 (k + 2) (- 38 k - 1) < 0 \Rightarrow 2 k^{2} + 3 k + 1 < 0 \Rightarrow (2 k + 1) (k + 1) < 0 \Rightarrow k \in (- 1, \frac{- 1}{2}) \end{matrix}$

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