if α, β are the roots of the quadratic polynomial p(x) = x^{2}-(k-6)x+2(2k-1).Find the value of k,ifα+β = 1/2αβ.

p(x) = x^{2} - (k + 6)x + 2(2k - 1)

Here a = 1, b = -(k + 6) and c = 2(2k - 1)

α + β = -b/a = -[-(k + 6)]/1 = k + 6

αβ = c/a = 2(2k - 1)/1 = 2(2k - 1)

Given,

α + β = 1/2* αβ

Putting the respective values of α + β and αβ, we have

k + 6 = 1/2*2(2k - 1)

k + 6 = 2k - 1

k = 7

So, value of k = 7

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