Multiple choice

The quadratic equation, which has the $A.M$ and $H.M$ of its roots as $\dfrac{3}{2}$ and $\dfrac{4}{3}$ respectively is:

  1. $x^{2}-3x+2=0$
  2. $x^{2}+3x+2=0$
  3. $x^{2}-3x-2=0$
  4. $x^{2}-3x+4=0$
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A Correct answer
Explanation

AM = (a+b)/2 = 3/2 => a+b = 3. HM = 2ab/(a+b) = 4/3 => 2ab/3 = 4/3 => ab = 2. The quadratic equation is x^2 - (sum of roots)x + (product of roots) = 0, which is x^2 - 3x + 2 = 0.

AI explanation

The arithmetic mean of the roots alpha and beta is given by (alpha + beta)/2 = 3/2, so the sum of the roots is alpha + beta = 3. The harmonic mean is 2*alpha*beta / (alpha + beta) = 4/3, and substituting the sum of 3 gives 2*alpha*beta / 3 = 4/3, meaning the product of the roots is alpha*beta = 2. A quadratic equation with roots alpha and beta is given by x^2 - (sum)x + (product) = 0. Substituting the calculated sum and product results in x^2 - 3x + 2 = 0.

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