Let , and be three distinct real roots of the equation x(3 x+2)^2+2=(a+12+9 x) x^2-b x+c where a, b , c R . If every solution of the inequality (x-a)^2(4 x+b)(x-c) 0 is also solution of the inequality 3 x^2+p x+p^2+6 p 0 then find number of integral values of ' p '

Step-by-step solution

aligned & x(3 x+2)^2+2=(a+12+9 x) x^2-b x+c & 4 x+2=a x^2-b x+c aligned which is quadratic in x but it satisfies by , and so it is an identity. a =0, ~b =-4, c =2 aligned & x^2(4 x-4)(x-2) 0 & x^2(x-1)(x-2) 0 & x (1,2) & 3 x^2+p x+p^2+6 p 0 satisfy by every x (1,2) & f(1) 0 and f(2) 0 & p^2+7 p+3 0 & and p^2+8 p+12 0 & p [ -7- 37 2 , -7+ 37 2 ] & (p+2)(p+6) 0 & p [-6,-2] & p [-6.5 ,-0 ]-6,-5,-4,-3,-2 aligned Number of integers are 5