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Let $f(x) = ax^2 + bx +c$, where $a, b$ and $c$ are certain constants and $a \neq 0$. It is known that $f(5) = -3 f(2)$ and that $3$ is a root of $f(x)=0$.

What is the value of $a+b+c?$

1. $9$
2. $14$
3. $13$
4. $37$
5. cannot be determined

f(x)=ax2+bx+c
f(5)=25a+5b+c
f(2)=4a+2b+c
25a+5b+c = -3(4a+2b+c)
25a+5b+c = -12a -6b -3c
37a+11b+4c=0
f(5)=-3f(2)

3 is a root, hence
f(3)=0,
9a+3b+c=0

37a+11b+4c=0 -------(i)
9a+3b+c=0 ----------(ii)

On solving (i) and (ii) we get, b = a, c = -12a  -----(iii)

Substituting (iii) in f(x) = 0
ax2 + ax – 12a = 0 => a(x2 + x – 12) = 0
Given that a ≠ 0 =>(x-3)(x+4) = 0 => x = 3 (or) -4
x=3 root is already given hence the other root is -4

but With the given two conditions (i) and (ii), it is not possible to find the value of a+b+c

34 points

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