Multiple choice

If AM and HM of the roots of a quadratic equation are 3/2 and 4/3 respectively, then that equation is.

  1. $x^2 + 3x + 2 = 0$
  2. $x^2 - 3x + 2 = 0$
  3. $x^2- 3x- 2 = 0$

  4. none of these
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B Correct answer
Explanation

AM = (r1+r2)/2 = 3/2 => r1+r2 = 3. HM = 2*r1*r2 / (r1+r2) = 4/3 => 2*r1*r2 / 3 = 4/3 => r1*r2 = 2. The quadratic equation is x^2 - (sum)x + (product) = 0, so x^2 - 3x + 2 = 0.

AI explanation

The arithmetic mean of the roots is the sum divided by 2, so the sum of the roots is 2 times 3/2, which equals 3. The harmonic mean is given by 4/3, and since the harmonic mean equals 2ab divided by (a plus b), we set 4/3 equal to 2 times the product divided by the sum 3. Solving this gives the product of the roots as 2. A quadratic equation with a sum of roots equal to 3 and a product equal to 2 is x^2 - 3x + 2 = 0.

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