If the roots of $x^2-2(7+3m)x+55m+45=0$ , are equal, then $m=$

The given equation is $x^2-2(7+3m)x+55m+45=0$

Given that this equation has equal roots

it implies that its discriminant is zero

Hence, $\Delta =b^2-4ac=0$

It implies that 

$4{\left(7+3m\right)}^2-4\left(55m+45\right)=0$

take common 4 both terms, and then simplify the above equation as below

$\begin{array}{rcl}9m^2+42m+49-55m-45& =& 0\\ 9m^2-13m+4& =& 0\end{array}$

Factorize this to get the values of $m$

Hence, 

$\begin{array}{rcl}9m^2-9m-4m+4& =& 0\\ 9m\left(m-1\right)-4\left(m-1\right)& =& 0\\ \left(m-1\right)\left(9m-4\right)& =& 0\end{array}$

it implies that 

$m=1,\frac{4}{9}$

Therefore, the given equation has equal roots when $m=\boxed{1,\frac{4}{9}}$