Question

Let $\alpha$ and $\beta$ be the two distinct roots of the equation of 2x2-6x+k=0, such that ($\alpha+\beta$) and $\alpha\beta$ are the distinct roots of the equation x2+px+p=0, then, the value of 8(k-p) ?

Answer 1

Given :
α and β are the distinct roots of the equation 2x2 - 6x + k = 0
⇒ αβ = $\frac{k}{2}$ …… ( Product of the roots )
⇒ α + β = $-(\frac{-6}{2})$ = 3 ( Sum of the roots )
So, (α + β) and αβ are the roots of the equation x2 + px + p = 0
⇒ α + β + αβ = -p
⇒ 3 + $\frac{k}{2}$ = -p …… (i)
⇒ (α + β)(αβ) = p
⇒ $3(\frac{k}{2})$ = p …… (ii)
Now , from eqn (i) and (ii) , we get
$3+\frac{k}{2}=-\frac{3k}{2}$
= 2k = -3
⇒ k = $-\frac{3}{2}$
By using the value of k , we get p
p = $\frac{3k}{2}=\frac{3}{2}(-\frac{3}{2})=-\frac{9}{4}$
Now , the value of 8(k-p) is
⇒ 8( k-p ) = $8(-\frac{3}{2}+\frac{9}{4})$
= -12 + 18
= 16
So, the correct answer is 16.