If the roots of the quadratic equation $x^{2}-ax+b=0$ are real and differ by a quantity less than $1$,then $\dfrac { { a }^{ 2 }-1 }{ 4 }

If the roots differ by a quantity less than 1, we can express them as (a/2 + d) and (a/2 - d), where d represents half of their difference. Because the difference is strictly between 0 and 1, the value of d squared must be between 0 and 1/4. The product of these roots equals a squared over 4 minus d squared. Substituting b for the product shows that b squared is greater than zero and less than 1/4, which rearranges to the given inequality.

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