Question

# lf $$ax^{2}-(2a+3)x +(3+5a)=0$$ has no real roots, then $$a$$ lies in the inverval

A
(34,34)
B
(,34)
C
(,34)
D
(,34)(34, )

Solution

## The correct option is D $$\left(-\infty, -\displaystyle \frac{3}{4}\right)\cup\left(\dfrac{3}{4},\ \infty\right)$$If $$ax^{2}+bx+c$$ has no real roots, then$$\Rightarrow b^{2}-4ac< 0$$$$\Rightarrow (2a+3)^{2}-4(a)(3+5a)< 0.$$$$\Rightarrow 4a^{2}+9+12a - 12a -20a^{2}< 0$$$$\Rightarrow 9-16a^{2}< 0$$$$\Rightarrow \left(-\infty, -\displaystyle \frac{3}{4}\right) \cup\left(\dfrac{3}{4},\ \infty\right)$$Mathematics

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