The quadratic equation ${\mathrm{x}}^2+\mathrm{ax}+\mathrm{b}=0$ (a, b real numbers) has both roots real and positive if and only if

We know that the quadraic equation $a{x}^2+bx+c=0$ has positive real roots if and only if  $b^2-4ac\ge 0$ and $ac>0$ and $ab<0$

Given that ${\mathrm{x}}^2+\mathrm{ax}+\mathrm{b}=0$ has both positive real roots

hence, $a^2-4b\ge 0$ and $a,b$are of different signs

Since $a^2-4b\ge 0\Rightarrow a\le -2\sqrt{b} or a\ge 2\sqrt{b}$

The value of $b$ is positive, hence $a$ must be negative

so that the required condition is $\boxed{a\le -2\sqrt{b},b is positive}$