Multiple choice

The condition that one root is twice the other root of the quadratic equation $\displaystyle x^{2}+px+q=0$ is

  1. $\displaystyle 2p^{2}=9q$
  2. $\displaystyle 3p^{2}=8q$
  3. $\displaystyle p^{2}=9q$
  4. $\displaystyle 4p^{2}=q$
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A Correct answer
Explanation

Roots are alpha and 2*alpha. Sum = 3*alpha = -p. Product = 2*alpha^2 = q. alpha = -p/3. 2*(-p/3)^2 = q => 2*p^2/9 = q => 2*p^2 = 9q.

AI explanation

Let the roots be m and 2m. The sum of the roots is m + 2m = -p, so m = -p/3. The product of the roots is m times 2m = 2m^2 = q. Substituting m = -p/3 into the product gives 2(-p/3)^2 = q, which simplifies to 2p^2/9 = q, or 2p^2 = 9q. The condition is 2p^2 = 9q.

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