If the roots of the equation x2 − bx + c = 0 are two consecutive integers,…

Concept

For any monic quadratic equation x2 − Sx + P = 0 with roots α and β, Vieta’s formulas give the sum of roots S = α + β and the product P = αβ. For a monic quadratic (leading coefficient 1), the discriminant equals the square of the difference between its roots: discriminant = (α − β)2.

Application

1. Let the two consecutive integer roots be n and n + 1.

2. By Vieta’s formulas for x2 − bx + c = 0: sum of roots b = n + (n + 1) = 2n + 1, and product of roots c = n(n + 1).

3. Substitute into b2 − 4c: (2n + 1)2 − 4n(n + 1).

4. Expand: (4n2 + 4n + 1) − (4n2 + 4n) = 1.

Cross-check

Using the difference-of-roots identity directly: the two roots differ by (n + 1) − n = 1, so the discriminant equals this difference squared, 12 = 1 — matching the algebraic expansion above.

So b2 − 4c = 1 for any pair of consecutive-integer roots.