If α and β are the zeros of the polynomial f(x) = $$kx^2+4x+4$$​ such that $$α^2+β^2= 24,$$​ then find the positive value of k.

Correct option is D

Given:

α and β are the zeros of the polynomial f(x) = $$kx^2 + 4x + 4 , α^2 + β^2 = 24$$​​

Formula Used:

​$$(a + b)^2 = a^2 + 2ab + b^2$$​​

Solution:

​$$=> α + β = \frac{- 4 }{ k} \\\ \\=> αβ = \frac{4 }{ k}\\\ \\=> (α + β)^2 = α^2 + β^2 + 2αβ\\\ \\=> (\frac{- 4}{ k})^2 = 24 +\frac{ 8 }{ k}\\\ \\=>\frac{ 16 }{ k^2} = 24 + \frac{8 }{k}\\\ \\=> \frac{16 }{ k^2} - \frac{8 }{k} = 24\\\ \\=> \frac{(16 - 8k) }{ k^2 }= 24\\\ \\=> 16 - 8k = 24k^2\\\ \\=> 24k^2 + 8k - 16 = 0\\\ \\=> 3k^2 + k - 2 = 0\\\ \\=> 3k^2 + 3k - 2k - 2 = 0\\\ \\=> 3k(k + 1) - 2(k + 1) = 0\\\ \\=> (3k - 2)(k + 1) = 0\\\ \\=> (3k - 2) = 0\\\ \\=> k = \frac{2 }{ 3}\\\ \\$$​​