Multiple choice

The quadratic equation whose roots are the $AM$ and $HM$ of the roots of the equation $\displaystyle x^{2}+7x-1=0$ is

  1. $\displaystyle 14x^{2}+14x-45=0$
  2. $\displaystyle 45x^{2}-14x+14=0$
  3. $\displaystyle 14x^{2}+45x-14=0$

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For x^2 + 7x - 1 = 0, let roots be a, b. a+b = -7, ab = -1. AM = (a+b)/2 = -3.5. HM = 2ab/(a+b) = 2(-1)/(-7) = 2/7. The equation is x^2 - (AM+HM)x + (AM*HM) = 0. AM+HM = -7/2 + 2/7 = (-49+4)/14 = -45/14. AM*HM = -7/2 * 2/7 = -1. Equation: x^2 + (45/14)x - 1 = 0, which is 14x^2 + 45x - 14 = 0.

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