Multiple choice

If the roots of the equation x2 - px + q = 0 differ by unity, then

  1. p2 = 4q

  2. p2 = 4q - 1

  3. p2 = 4q + 1

  4. p2 = 2q
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C Correct answer
Explanation

If roots are r and r+1, their sum is 2r+1 = p and their product is r(r+1) = q. Substituting r = (p-1)/2 into the product equation gives ((p-1)/2)((p+1)/2) = q, which simplifies to (p^2-1)/4 = q, or p^2 = 4q + 1.

AI explanation

Let the roots of the equation x2 - px + q = 0 be alpha and beta. From the relation between roots and coefficients, the sum of the roots alpha + beta equals p, and the product alpha * beta equals q. The identity for the square of a difference states that (alpha - beta)2 = (alpha + beta)2 - 4*alpha*beta. Substituting the known values and the given difference of one gives 12 = p2 - 4q. Rearranging this equation yields p2 = 4q + 1.

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