The value of a for which the difference of the roots of the equation $\mathrm{a}\mathrm{x}^{2}+(\mathrm{a}-1)\mathrm{x}+2=0$ is minimum is given by

The difference between the roots of the quadratic equation ax squared plus (a minus 1)x plus 2 equals 0 is given by the square root of the discriminant divided by the absolute value of a, which is the square root of ((a minus 1) squared minus 8a) divided by a. Simplifying the expression under the square root gives the square root of (a squared minus 10a plus 1) divided by a. To minimize the difference, we minimize the expression (a squared minus 10a plus 1) divided by a squared, which simplifies to 1 minus (10 divided by a) plus (1 divided by a squared). Differentiating this with respect to a and setting it to zero yields 10 divided by a squared minus 2 divided by a cubed equals 0, which solves to a equals one fifth.

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