Question

The quadratic equation whose roots are the am and hm between the roots of the equation 2x2 - 3x+5=0

Answer 1

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