There is a standard way of solving for $Q_1$ and $Q_2$.
Determine the profit functions.
Determine the best response function for the firms.
Substitute $Q_1$ or $Q_2$ in the other profit function and solve.
All these steps are already mentioned, so you know what to do. Below you can search for your mistake.
The profit function for firm 1 equals $\Pi_1= P_1Q_1-C_1=Q_1 \cdot (100-0.5(Q_1+Q_2)) - 50Q_1$
The profit function for firm 2 equals $\Pi_2=P_2Q_2-C_2=Q_2 \cdot (100-0.5(Q_1+Q_2)) - 24Q_2$
The best response function can be determined by deriving the profit function of firm 1 w.r.t. $Q_1$ and for firm 2 w.r.t. $Q_2$ and set them equal to zero
$$\frac{\partial \Pi_1}{\partial Q_1}=100-Q_1-0.5Q_2-50=50-Q_1-0.5Q_2=0$$
$$\implies Q_1=50-0.5Q_2$$
$$\frac{\partial \Pi_2}{\partial Q_2}=100-Q_2-0.5Q_1-24=76-Q_2-0.5Q_1=0$$
Now we can make the substitution
$$76-Q_2-0.5 \cdot (50-0.5Q_2)=0$$
$$\implies 51-Q_2+0.25Q_2=0 \implies 0.75Q_2=51$$
And thus we find $Q_2=68$ and can solve easily for $Q_1$
$$Q_2=68 \ \text{and} \ Q_1=50-0.5 \cdot 68=16$$
Since there are two quantities you can set independently this is a two-variable calculus problem. You should call the quantity produced by the first process $q_1$ and the second quantity $q_2$ and then write an expression $P(q_1,q_2)$ for the profit. To find the local extrema of the profit, you take the partial derivative with respect to each parameter and set both of them equal to zero. This will give you two equations in two variables to solve.
It seems a bit odd that it would be a multivariate calculus problem given the prerequisites and background, but that's my best interpretation of the problem.
Best Answer
Hint: In a competitive market the firms produce on the long run at a level where the average total cost function has its minimum. Thus you have to find the minimum of
$$\frac{TC(Q)}{Q}=4Q+100+\frac{100}{Q}$$
The minimum can be found by setting the derivative equal to $0$. In this case it is good to remember that $Q$ is defined for non-negative values only. The picture below shows the course of the average cost function.
The price of the product will be equal the solution.