Multiple choice

If the roots of equation $x^2 + px + q = 0$ differ by 1, then

  1. $p^2 = 4q$
  2. $p^2 = 4q + 1$
  3. $p^2 = 4q - 1$

  4. None of these
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B Correct answer
Explanation

Let roots be alpha and beta. alpha - beta = 1. (alpha - beta)^2 = 1. (alpha + beta)^2 - 4*alpha*beta = 1. (-p)^2 - 4q = 1. p^2 = 4q + 1.

AI explanation

Let the roots be alpha and alpha plus one, so their difference is one. The sum of the roots gives p equals negative two alpha minus one, and the product gives q equals alpha squared plus alpha. Substituting alpha equals the quantity negative p minus one divided by two into the product equation yields q equals p squared minus one divided by four, which rearranges to p squared equals four q plus one.

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