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 distinct roots of 2x² - 6x + k = 0.
Product of roots: αβ = k/2 …… (1)
Sum of roots: α + β = -(-6)/2 = 3 …… (2)
The roots of x² + px + p = 0 are (α + β) and αβ.
Sum of these roots: (α + β) + αβ = -p
Substituting (1) and (2): 3 + k/2 = -p …… (3)
Product of these roots: (α + β)(αβ) = p
Substituting (1) and (2): 3(k/2) = p …… (4)
Equating (3) and (4):
3 + k/2 = -(3k/2)
6 + k = -3k
4k = -6
k = -6/4 = -3/2
Substitute the value of k into (4) to find p:
p = (3/2) * (-3/2) = -9/4
Calculate 8(k - p):
8(k - p) = 8(-3/2 - (-9/4))
= 8(-3/2 + 9/4)
= 8(-6/4 + 9/4)
= 8(3/4)
= 6
The final answer is 6.