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

$\because A=\frac{a+b+c}{3}\Rightarrow a+b+c=3A\to \left(1\right)$

$\because G={\left(abc\right)}^{\frac{1}{3}}\Rightarrow abc=G^3\to \left(2\right)$

$\because \frac{1}{H}=\frac{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}{3}\Rightarrow ab+bc+ca=\frac{3G^3}{H}\to \left(3\right)$

  $\because$ Equation with roots a,b,c is  $x^3-s_1{x}^2+s_{2}x-s_3=0$

$\Rightarrow x^3-\left(a+b+c\right)x^2+\left(ab+bc+ca\right)x-\left(abc\right)=0$.