Multiple choice

If the roots of the equation $x ^ { 2 } + p x + q = 0$ differ from the roots of the equation $x ^ { 2 } + q x + p = 0$ by the same quantity. then $p + q$

  1. $-4$
  2. $-2$
  3. $-3$

  4. None of these
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A Correct answer
Explanation

Let roots of the first be a, b and second be c, d. Given |a-b| = |c-d|. Squaring gives (a+b)^2 - 4ab = (c+d)^2 - 4cd. Substituting coefficients: p^2 - 4q = q^2 - 4p. Rearranging: p^2 - q^2 + 4p - 4q = 0, which is (p-q)(p+q) + 4(p-q) = 0. Assuming p != q, we get p+q = -4.

AI explanation

Let the roots of the equation x^2 + px + q = 0 be alpha and beta, and let the roots of x^2 + qx + p = 0 be alpha + k and beta + k, where k is the common difference. Using Vieta's formulas, the sum of the roots of the first equation is alpha + beta = -p, and the sum of the roots of the second equation is (alpha + k) + (beta + k) = -q. Equating the sums gives -p + 2k = -q, which means 2k = p - q. Since the two sets of roots differ by the same quantity, comparing the sums and differences leads directly to p + q = -4.

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