Multiple choice

If difference of roots of the equation $\displaystyle x^{2}+px+8= 0$ is $2$, then $p$ is equal to

  1. $\displaystyle \pm 2$
  2. $\displaystyle \pm 4$
  3. $\displaystyle \pm 6$
  4. $\displaystyle \pm 8$
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C Correct answer
Explanation

Let roots be alpha and beta. alpha + beta = -p, alpha * beta = 8. (alpha - beta)^2 = (alpha + beta)^2 - 4*alpha*beta. 2^2 = (-p)^2 - 4(8). 4 = p^2 - 32. p^2 = 36, so p = +/- 6.

AI explanation

Let the roots be alpha and beta. By Vieta's formulas, the sum of the roots is alpha + beta = -p and the product is alpha*beta = 8. The difference of the roots formula is given by |alpha - beta| = sqrt(Discriminant) = sqrt(p^2 - 4ac). Substituting the values gives sqrt(p^2 - 32) = 2. Squaring both sides results in p^2 - 32 = 4, so p^2 = 36, which means p is equal to plus or minus 6.

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