Given,

$x^2-7x+3c=0$ and $x^2+x-5c=0$ have a common root,

So assuming $\alpha$ to be common root then,

${\alpha }^2-7\alpha +3c=0 ...\left(1\right)$

${\alpha }^2+\alpha -5c=0 ......\left(2\right)$

Solving equation $\left(1\right) \& \left(2\right)$ we get,

$\frac{{\alpha }^2}{35c-3c}=\frac{-\alpha }{-5c-3c}=\frac{1}{1+7}$

$\Rightarrow \alpha =c \& {\alpha }^2=4c$

$\Rightarrow c^2=4c$

$\Rightarrow c=0 \text{or} c=4$

So the value of $c=4$

Now solving $x^2-3x+c=x^2-3x+4$

So the value of discriminant $D=9-16<0$

So the value of $x^2-3x+c$ is always positive.