Multiple choice

If the roots of the equation $\displaystyle x^{2}-bx+c=0 $ be two consecutive integers, then $\displaystyle b^{2}-4c $ equals

  1. $3$
  2. $-2$
  3. $1$
  4. $2$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If the roots are n and n+1, their sum is 2n+1 = b and their product is n(n+1) = c. The discriminant b^2 - 4c = (2n+1)^2 - 4(n^2+n) = 4n^2 + 4n + 1 - 4n^2 - 4n = 1.

AI explanation

Let the two consecutive integer roots be n and n+1. The sum of the roots is n + (n + 1) = 2n + 1, which equals b, and the product of the roots is n(n + 1) = c. We substitute these into the expression b^2 - 4c to get (2n + 1)^2 - 4n(n + 1). Expanding the square gives 4n^2 + 4n + 1 - 4n^2 - 4n, which simplifies to 1.

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