If the roots of the equation $\displaystyle 4x^{2}+5k =(5k+1)x$ differ by unity, then the negative value of $k$ is
- $-3$
- $\displaystyle -\frac{1}{5}$
- $\displaystyle -\frac{3}{5}$
- $None\:of\:these$
Equation: 4x^2 - (5k+1)x + 5k = 0. Roots differ by 1, so |alpha - beta| = 1. (alpha - beta)^2 = 1, which is (alpha+beta)^2 - 4*alpha*beta = 1. ((5k+1)/4)^2 - 4*(5k/4) = 1. Solving (5k+1)^2 / 16 - 5k = 1 leads to 25k^2 + 10k + 1 - 80k = 16, 25k^2 - 70k - 15 = 0. Dividing by 5 gives 5k^2 - 14k - 3 = 0. Factoring: (5k+1)(k-3) = 0. Negative value is -1/5.
Rearrange the given equation into the standard quadratic form 4x^2 - (5k+1)x + 5k = 0. If the roots differ by unity, the relationship between the roots gives (alpha - beta)^2 = 1. Using the identity (alpha - beta)^2 = (alpha + beta)^2 - 4(alpha)(beta), we substitute the sum and product of the roots to get [ (5k+1)/4 ]^2 - 4(5k/4) = 1. This simplifies to 25k^2 + 10k + 1 - 80k = 16, which further reduces to the quadratic equation 25k^2 - 70k - 15 = 0. Dividing the entire equation by 5 gives 5k^2 - 14k - 3 = 0, and factoring this yields (5k + 1)(k - 3) = 0. The two possible values for k are -1/5 and 3, so the required negative value is -1/5.