The first thing to do is to find the roots.
The sum of the roots, x1 + x2 = -b / a
x1 + x2 = 3 / 2
The product of the roots, x1 * x2 = c / a
x1 * x2 = 5 / 2
The AM means arithmetic mean and is calculated using the formula:
AM = (x1 + x2) / 2
AM = (3 / 2) / 2
AM = 3 / 4 = y1
The HM means harmonic mean and is calculated using the formula:
2 / HM = 1 / x1 + 1 / x2
or
2 / HM = (x1 + x2) / x1 * x2
Rewriting in terms of HM alone on the left side:
HM = 2 x1*x2 / (x1 + x2)
HM = 2 (5 / 2) / (3 / 2)
HM = 10 / 3 = y2
Creating an equation from y1 and y2:
(y – ¾) (y – 10/3) = 0
y^2 – 10/3 y – ¾ y + 30/12 = 0
y^2 - 49/12 y + 30/12 = 0 (ANSWER)
or
y^2 – 4.083y + 2.5 = 0