If x₁ and x₂ are the arithmetic and harmonic means of the roots of the equation ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0$, the equation whose quadratic roots are x₁ and  x₂, is

Let $\alpha$ and $\mathrm{\beta }$ be the roots of  ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0$

$\begin{array}{rl} & {\mathrm{x}}_1=\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}=-\frac{\mathrm{b}}{2\mathrm{a}}\\ & {\mathrm{x}}_2=\frac{2\mathrm{\alpha \beta }}{\mathrm{\alpha }+\mathrm{\beta }}=\frac{2\cdot \frac{\mathrm{c}}{\mathrm{a}}}{-\frac{\mathrm{b}}{\mathrm{a}}}=-\frac{2\mathrm{c}}{\mathrm{b}}\end{array}$

The required equation is

$\begin{array}{r}{\mathrm{x}}^2-[(-\frac{\mathrm{b}}{2\mathrm{a}})+(-\frac{2\mathrm{c}}{\mathrm{b}})]\mathrm{x}+\frac{2\mathrm{bc}}{2\mathrm{ab}}=0\\ 2{\mathrm{abx}}^2+({\mathrm{b}}^2+4\mathrm{ac})\mathrm{x}+2\mathrm{bc}=0\end{array}$