Question

If A, G & H are respectively the A.M., G.M. & H.M. of three positive numbers a, b, & c, then equation whose roots are a, b, & c is given by

Answer 1

`A = (a+b+c)/3`
`a+b+c = 3A `eqn(1)
`G= (abc)^(1/3)`
`abc = G^3`
`H= 3/((1/a) + (1/b) + (1/c))`
`H= 3/((bc+ab+ca)/(abc)) = (3abc)/(b+bc+ ca)`
`ab+bc+ca = (3abc)/H = (3G^3)/H` eqn(3)
`x^3 - (a+b+c)x^2 + (ab+bc+ca)x - (abc) = 0`
`x^3 - 3Ax^2 +(3G^3/H)x - G^3 = 0`
`x^3 - 3Ax^2 +(3G^3/H)x - G^3 = 0`
answer