jmc

We can factor the equation as $$x^6(x^4+7x^3+14x^2+1729x-1379)=0.$$Since we are looking for positive real solutions, this reduces to $$x^4+7x^3+14x^2+1729x-1379.$$Consider the function $f(x)=x^4+7x^3+14x^2+1729x-1379.$ This is increasing for $x>0.$ Also, $f(0)<0$ and $f(1)>0,$ so there is exactly $\boxed{1}$ positive real solution, which lies in the interval $(0,1).$