Find $k$ in the equation $5 x^{2} - k x + 1 = 0$ such that the difference between the roots of the equation is unity.

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Solution

${5 x}^{2} - k x + 1 = 0 l e t r o o t s b e x_{1} & x_{2} s u m o f r o o t s = x_{1} + x_{2} = \frac{1}{5} x_{2} - x_{1} = 1 {(x_{1} + x_{2})}^{2} = {(x_{2} - x_{1})}^{2} + 4 x_{1} x_{2} \frac{k^{2}}{25} = 1 + \frac{4}{5} k^{2} = \frac{9}{5} * 25 k^{2} = 45 k = \pm \sqrt{45} = \pm 3 \sqrt{5}$

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