If the roots of $4 x^{2} - (5 k + 1) x + 5 k = 0$ differ by unity then the sum of all the possible values of $k$ is

\frac{11}{5}

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\frac{12}{5}

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\frac{13}{5}

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\frac{14}{5}

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Solution

The correct option is D $\frac{14}{5}$
Let $α$ and $β$ be the roots the given equation.
$\Rightarrow α + β = \frac{5 k + 1}{4}, α \times β = \frac{5 k}{4} \Rightarrow | α - β | = 1$
$\Rightarrow (α - β)^{2} = (α + β)^{2} - 4 α β = 1 \Rightarrow 3 = 5 k^{2} - 14 k \Rightarrow k = 3, \frac{- 1}{5}$
$\Rightarrow$ Sum of possible values of $k = \frac{14}{5}$

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