Question

Let $y=f(x)$ be a quadratic polynomial with leading co-efficient 1 . Let $\alpha$ and $\beta$ be the roots of the equation $f(x)=0$ and $\gamma$ and $\delta$ be the roots of the equation $f(x)=8 x-32$. If $\alpha, \beta, \gamma$ and $\delta$ (in order) are in A.P. Then find the value of $f (4)$.

Answer 1

Let $f(x)=x^2-P x+q, \alpha=a, \beta=a+d$
$a + a + d = P $
$a +2 d + a +3 d = P +\delta$
$\Rightarrow d =2 $
$a ( a +2)= q$
$( a +4)( a +6)= q +32$
$\Rightarrow 8 a +24=32 \Rightarrow a =1 $
$\Rightarrow q =3 \text { and } P =4 $
$f ( x )= x ^2-4 x +3 $
$f (4)=3$