Let α and β be the two distinct roots of the equation 2x^2-6x+k=0, such that (α+β) and αβ are the distinct roots of the equation x^2+px+p=0. Then, the value of 8(k-p) is

Options

1. 2
2. 6
3. 9
4. none

Correct Answer

6

Explanation

Let α and β be the roots of the quadratic: 2x² − 6x + k = 0 Sum of roots (α + β) = 6/2 = 3 Product of roots (αβ) = k/2 Given: (α + β) and (αβ) are the roots of: x² + p x + p = 0 So the roots of this equation are 3 and k/2. For x² + p x + p = 0: Sum of roots = 3 + k/2 = −p Product of roots = 3 × (k/2) = 3k/2 = p From product: p = 3k/2 Substitute this in the sum equation 3 + k/2 = −3k/2 Multiply both sides by 2: 6 + k = −3k 4k = −6 k = −3/2 Then p = 3k/2 = 3(−3/2)/2 = −9/4 Compute 8(k − p): 8(−3/2 − (−9/4)) = 8(−6/4 + 9/4) = 8(3/4) = 6 Final Answer: 6

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