Let α and β be two distinct root of the equation 2x2 – 6x + k = 0, such that (α + β) and αβ are the two roots of the equation x2 + px + p = 0. Then the value of 8(k - p)?
Explanation:
α and β be two distinct root of 2x2 – 6x + k = 0, ∴ α + β = -(-6)/2 = 3 And, α × β = k/2 = k/2
Now, 3 and k are the roots of the equation x2 + px + p = 0. ∴ Sum of the roots = 3 + k/2 = -(p)/1 = -p …(1) ∴ Product of the roots = 3 × k/2 = (p)/1 = p …(2)
(1) + (2) ⇒ 3 + k/2 + 3k/2 = p – p = 0 ⇒ k = -3/2
⇒ p = -9/4 [from (2)]
Now, we need to find 8(k - p) = 8-32--94 = 834 = 6
Hence, 6.
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