If one zero of a polynomial 3x^{2}-8x+2k+1 is seven times the other. Find the value of k

(1) Let f(x) = 3x^{2}+ 8x + (2k + 1) and α and β be its zeroes

Here a = 3, b = 8 and c = 2k + 1

Given,

α = 7β -(i)

Sum of roots, α + β = -b/a

7β + β = -8/3 [Using (i)]

8β = -8/3

β = -1/3

Putting β = -1/3 in (i), we have

α = 7*-1/3 = -7/3

So, the zeroes are α = -7/3 and β = -1/3

Now,

Product of roots = -1/3*-7/3

c/a = 7/9

(2k + 1)/3 = 7/9

2k + 1 = 7/3

2k = 4/3

k = 2/3

So, value of k = 2/3

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