Question

Let $f (x) = px^2 + qx + r$ , where $p, q, r$ are constants and $p \neq 0$ . If $f (5) = -3 f (2)$ and $f (-4) = 0$ , then the other root of $f$ is

• A. 3
• B. -7
• C. -2
• D. 2
• E. 6

Answer 1

Answer: (A)

$f(x)=p x^{2}+q x +r$
$f(-4)=0$
$\Rightarrow \, 16 p-4 q +r=0$ ...(i)
One root is $x=-4$
and $f(5)=-3 f(2)$
$25 p+5 q +r=-3(4 p+2 q +r)$
$\Rightarrow 37 p+11 q+4 r=0$ ...(ii)
Eq. (ii) - Eq. (i), we get
$\Rightarrow -27 p+27 q =0$
$\Rightarrow p =0$
Then, equation is $p x^{2}+q x +r=0$
Roots $=-4, \alpha$
Sum of roots $=-4 x+\alpha=-\frac{p}{q}=-1$
So, another root $\alpha=3$.