Multiple choice

If one root of the quadratic equation $\displaystyle ax^{2}+bx+c=0$ is double the other then which one of the following is correct

  1. $\displaystyle b^{2}=3ac$
  2. $\displaystyle 2b^{2}=5ac$
  3. $\displaystyle 2b^{2}=9ac$
  4. $\displaystyle 2b^{2}>9ac$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let roots be alpha and 2*alpha. Sum = 3*alpha = -b/a, so alpha = -b/(3a). Product = 2*alpha^2 = c/a. Substituting alpha: 2*(-b/(3a))^2 = c/a => 2*(b^2 / 9a^2) = c/a => 2*b^2 = 9ac.

AI explanation

Let the roots be m and 2m. The sum of the roots is 3m = -b/a, so m = -b/(3a). The product of the roots is 2m^2 = c/a. Substituting m into the product gives 2(-b/(3a))^2 = c/a, which simplifies to 2b^2/(9a^2) = c/a. Multiplying both sides by 9a^2 results in 2b^2 = 9ac. The correct relation is 2b^2 = 9ac.

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