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1)find the quadratic polynomial with zeroes 3+√2 & 3-√2. 2)if alpha*beta are zeroes of x^2 +7x+12,then find the value of (1/2)alpha +(1/2)beta-2(alpha*beta). 3)if alpha ,beta are the ratio of quadratic polynomial p(x)=x^2-(k-6)x+2(2k-1). find the value of k ,if alpha +beta =(1/2alpha*beta).

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1) sum of roots = 3 + √2 + 3 - √2 = 6 product of roots = (3+√2) (3-√2) = 3² - 2 = 7 quadratic polynomial = x² - sum of roots + product of roots x² - 6 x + 7

2) α + β = - coefficient of x / coefficient of x² = -7 α β = constant term / coefficient of x² = 12 1/2 α + 1/2 β - 2(α β) = 1/2 (α + β) - 2 αβ = -7/2 - 2 * 12 = -55/2

3) x² - (k-6) x + 2 (2k-1) α + β = k - 6 α β = 4 k - 2

as α + β = 1/ (2αβ) , k - 6 = 1/(8 k -4)

8 k² - 4 k - 48 k + 24 = 1 8 k² - 52 k + 23 = 0 factors of 23 * 8 = 23 * 2 * 6 = 46 * 6 whose sum is 52. 8k² - 4 k - 46 k + 23 = 0 4 k (2 k - 1) - 23 (2k - 1) = 0 (4k - 23) ( 2k - 1) = 0