As shown in the figure, charge 1 is \(3.94 \mu \mathrm{C}\) and is located at \(x_{1}=-4.7 \mathrm{~m},\) and charge 2 is \(6.14 \mu \mathrm{C}\) and is at \(x_{2}=\) \(12.2 \mathrm{~m} .\) What is the \(x\) -coordinate of the point at which the net force on a point charge of \(0.300 \mu \mathrm{C}\) is zero?

The process of mastering physics concepts often begins with understanding the fundamental forces that govern interactions in the universe. Among these, the electric force is a key player. It's described by Coulomb's Law, which states that the electric force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

The formula derived from Coulomb's Law is thus:
\[ F = k \frac{q_1q_2}{r^2} \]
where \(k\) represents Coulomb's constant, \(q_1\) and \(q_2\) are the magnitudes of the charges, and \(r\) is the distance between the charges. This relationship is pivotal for solving problems involving electric forces and can be visualized by imagining charges exerting invisible lines of force upon each other, attracting or repelling depending on their polarities. In the context of our problem, we use this law to assess the forces acting on a nominal third charge to determine where it would experience no net force.

Another intriguing scenario in the realm of physics is the concept of net force zero. This occurs when all the forces acting upon an object balance out, resulting in no overall force impacting the object's motion. It's an essential condition for equilibrium in various physical systems.

In the given exercise, we seek the specific point at which a third charge, introduced into the system of two existing charges, experiences equal and opposite forces from these charges, leading to zero net force. To clarify:
\[ F_{13} = -F_{23} \]
This condition indicates that the force due to charge 1 on this third charge (\(F_{13}\)) must be equal in magnitude but opposite in direction to the force due to charge 2 (\(F_{23}\)). Balancing these forces is similar to finding the 'sweet spot' in a tug of war where neither side gains ground—the spot where the third charge 'feels' both forces equally but remains motionless due to their cancelation.

Complex interactions often lead to equations that can't be solved through simple algebra, which is when the quadratic equation comes into play. It is an indispensable tool for solving second-order polynomials of the form:
\[ ax^2 + bx + c = 0 \]
To find the roots of such an equation, which represent the solutions to our problem, the quadratic formula is used:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our exercise, after accounting for Coulomb's Law and setting up an equation for net force zero, we arrive at a quadratic equation that we solve to find potential positions for the third charge. However, not all solutions may make physical sense. Hence, as detailed in the solution steps, one must scrutinize the context to select the appropriate physical solution.