Multiple choice

The roots of the equation $\displaystyle x^{2}-px+q=0$ are consecutive integers. Find the discriminate of the equation.

  1. $-1$
  2. $0$
  3. $1$

  4. None of these
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C Correct answer
Explanation

If the roots are consecutive integers, their difference is 1. The discriminant equals the square of the difference between the roots, so it is 1.

AI explanation

Let the two consecutive integer roots be n and n + 1. By Vieta's formulas, the sum of the roots is p = n + (n + 1) = 2n + 1. The discriminant of a quadratic equation is given by D = b^2 - 4ac, which for this equation is p^2 - 4q. Substituting q = n(n + 1) and p = 2n + 1, we find D = (2n + 1)^2 - 4(n^2 + n) = 4n^2 + 4n + 1 - 4n^2 - 4n = 1. The discriminant is 1.

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