Multiple choice

If the absolute difference between two roots of the equation $x^2 + px + 3 = 0$ is $\sqrt{p}$, then $p$ equals

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

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Roots r1, r2 satisfy r1+r2 = -p and r1*r2 = 3. Difference |r1-r2| = sqrt((r1+r2)^2 - 4r1r2) = sqrt(p^2 - 12). Given sqrt(p^2 - 12) = sqrt(p), so p^2 - 12 = p. p^2 - p - 12 = 0. (p-4)(p+3) = 0. p=4 or p=-3. Since sqrt(p) must be real, p=4.

AI explanation

Let the roots be alpha and beta, so their absolute difference is given by the square root of ((alpha + beta)^2 - 4*alpha*beta). By Vieta's formulas for x^2 + px + 3 = 0, the sum of the roots is -p and the product is 3. Setting the difference to sqrt(p) gives the equation sqrt(p^2 - 12) = sqrt(p), which simplifies to p^2 - p - 12 = 0. Solving this quadratic equation yields p = 4 or p = -3, but since p must be non-negative to have a real square root, p equals 4.

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