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If $a x^{2} - (2 a + 3) x + (3 + 5 a) = 0$ has no real roots then ‘a’ lies in the interval

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$(- \infty, - \frac{3}{4})$

$(- \infty, \frac{3}{4})$

$(- \frac{3}{4}, \frac{3}{4})$

$(- \infty, - \frac{3}{4}) U (\frac{3}{4}, \infty)$

answer is D.

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Detailed Solution

$\begin{array}{l} Δ < 0 \Rightarrow (2 a + 3)^{2} - 4 a (3 + 5 a) < 0 \\ \Rightarrow 4 a^{2} + 12 a + 9 - 12 a - 20 a^{2} < 0 \\ ∴ a \in (- \infty, - \frac{3}{4}) U (\frac{3}{4}, \infty) \end{array}$

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