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