If α, β are the roots of 9x² - 27 x + k = 0, find the value of k such that 2α + 5 β = 7?
- 15
- 9
- 8
- None of these
Explanation
Given quadratic equation: 9x² - 27x + k = 0
Step 1: Find the sum and product of roots
Sum of roots (α + β) = -(-27)/9 = 3
Product of roots (αβ) = k/9
Step 2: Use the given condition
2α + 5β = 7
We also know α + β = 3
Step 3: Express α in terms of β or vice versa
From α + β = 3, α = 3 - β
Step 4: Substitute α in the given condition
2(3 - β) + 5β = 7
6 - 2β + 5β = 7
3β = 1
β = 1/3
Step 5: Find α
α = 3 - β = 3 - 1/3 = 8/3
Step 6: Find k using the product of roots
αβ = k/9
(8/3)(1/3) = k/9
8/9 = k/9
k = 8
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