Given-that-the-solution-set-of-the-quadratic-inequality-ax-2-bx-c-gt-0-is-2-3-Then-the-solution-set-of-the-inequality-cx-2-bx-a-lt-0-will-be-

Question Number 162366 by cortano last updated on 29/Dec/21

G i v e n t h a t t h e s o l u t i o n s e t o f t h e

q u a d r a t i c i n e q u a l i t y {a x}^{2} + b x + c > 0

i s (2, 3) . T h e n t h e s o l u t i o n s e t

o f t h e i n e q u a l i t y {c x}^{2} + b x + a < 0

w i l l b e

Answered by mr W last updated on 29/Dec/21

a < 0

x^{2} + \frac{b}{a} x + \frac{c}{a} = (x - 2) (x - 3)

\frac{b}{a} = - (2 + 3) = - 5 \Rightarrow b = - 5 a

\frac{c}{a} = 2 \times 3 = 6 \Rightarrow c = 6 a

{c x}^{2} + b x + a = a (6 x^{2} - 5 x + 1) < 0

s i n c e a < 0,

\Rightarrow 6 x^{2} - 5 x + 1 > 0

(2 x - 1) (3 x - 1) > 0

\Rightarrow x < \frac{1}{3} o r x > \frac{1}{2}

i . e . x \in (- \infty, \frac{1}{3}) \land (\frac{1}{2}, + \infty)

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