Multiple choice

The condition for equation ${x}^{2}-bx+c=0$ to have two consecutive integers as its roots is

  1. ${b}^{2}-{c}^{2}=1$
  2. ${b}^{2}-4c=1$
  3. ${c}^{2}-4{b}^{2}=1$
  4. $b=c$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If roots are k and k+1, their sum is 2k+1 = b and product is k(k+1) = c. Then b^2 - 4c = (2k+1)^2 - 4(k^2+k) = 4k^2 + 4k + 1 - 4k^2 - 4k = 1.

AI explanation

Let the two consecutive integer roots be k and k+1. Their sum is 2k + 1 and their product is k squared + k. Equating these to the coefficients of the quadratic x squared - b x + c = 0 gives b = 2k + 1 and c = k squared + k. Substituting these expressions into the condition for real roots, we evaluate b squared - 4c, which becomes (2k + 1) squared - 4(k squared + k) and simplifies exactly to 1.

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