Sum.of all values of X for which this equation is solved.

Let T^2 = X.

The equation to solve can be rewritten as

T^2 + T - 6 = 0. or (T+3)*(T-2) = 0.

T = -3 or T = 2 solves the equation above.

So

X = 9 or X = 4 . Hence the sum.of all the solutions for which this equation holds true is 13

$\begin{aligned} 6 & = X + \sqrt{X} \\ \Rightarrow (6 - X)^2 & = \sqrt{(X)^2} ~~~~~[\text{Square both sides}] \\ \Rightarrow 6^2 - 12X + X^2 & = X \\ \Rightarrow X^2 - 13X + 36 & = 0 \\ \Rightarrow X^2 - 9X - 4X + 36 & = 0 \\ \Rightarrow X(X - 9) - 4(X - 9) & = 0 \\ \Rightarrow (X - 9) (X - 4) &= 0 \\ \end{aligned}$ .

We get $X = 4$ and $X = 9$ . The answer is $4 + 9 = 13.$