Question

Let a, ß and y be three distinct real roots of the equation x(3x + 2)2 + 2 = (a + 12 + 9x)x2 – bx + c where a, b, c E R. If every solution of the inequality (x – a)2 (4x + b)(x – c) < 0 is also solution of the inequality 3x2 + px + p2 + 6p < 0 then the number of integral values of 'p' is​

Answer 1

α & β are the roots of the equation x

2

−x+p

γ & δ are the roots of the equation x

2

−4x+q

Since, α,β,γ,δ are in G.P

Let α=a, β=ar, γ=ar

2

, δ=ar

3

α+β=a+ar=1 [∵ Sum of the roots =1] ...(1)

γ+δ=ar

2

+ar

3

=r

2

(a+ar)=4 [∵ Sum of the roots =4] ...(2)

From (1) & (2), we get r

2

=4⇒r=±2

Substituting r in (1), we get a=−1,

3

1

Now, αβ=p=a

2

r=−2,

9

2

and γδ=q=a

2

r

5

=−32,

9

32