Quadratic Equations-3 (CAT-2008)

Directions for Questions 1 and 2: 

Let f(x) = ax² + bx + c, where a, b and c are certain constants and a ≠ 0. It is known that f(5) = −3f(2) and that 3 is a root of f(x) = 0.

1. What is the other root of f(x) = 0?

(1)   −7        (2)   − 4         (3)     2         (4)   6        (5)   cannot be determined

2. What is the value of a + b + c?      

(1)  9          (2)   14          (3)     13          (4)   37     (5)   cannot be determined

Solution follows here:

Solution:

f(x) = ax² + bx + c

f(5) = −3f(2) => 25a+5b+c = -3(4a+2b+c) => 37a+11b+4c = 0    ---(1)

3 is a root of f(x) = 0 => f(3) = 0 => 9a+3b+c = 0                          ---(2)

On solving (1) and (2) we get, b = a, c = -12a                                ---(3)

Substituting (3) 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

Answer option for first question is (2)

With the given two conditions (1) and (2), it is not possible to find the value of a+b+c

Hence answer option for second question is (5)