Paragraph: Let f(x)=a x^2+b x+c , where a, b, c R and a 0 . It is known that f(5)=-3 f(2) and that 3 is a root of f(x)=0 . Question: The value of the sum (a+b+c) is

Step-by-step solution

aligned & f(x)=a x^2+b x+c & f(5)=-3 f(2) and x=3 satisfies f(x)=0 & (i) 9 a+3 b+c=0 ......(1) & 25 a+5 b+c=-3(4 a+2 b+c) & 37 a+11 b+4 c=0 ......(2) aligned Also, 36 a+12 b+4 c=0 ......(3) [multiplying equation (1) by 4] aligned & a - b =0 & a = b & + =- b a =-1, but =3 & 3+ =-1 =-4 aligned hence the other roots is -4 (ii) a+b+c=f(1) now f(x)=a [x^2- (- b a ) x+ c a ] aligned & =a[(x- )(x- )] & =a[(x-3)(x+4)]=a (x^2+x-12 ) & f(1)=-10 a aligned can not be uniquely determined (iii) f^ (x)=2 x+11=0 x=- 1 2 Alternativ