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.