Final answer:
To determine the character of complex number solutions without solving, identify unknowns and knowns, use appropriate equations to relate them, and substitute known values to check the solutions' reasonableness, focusing on the discriminant in the case of quadratic equations.
Step-by-step explanation:
To determine the character of the solutions of the equation in the complex number system without actually solving it, we need to undertake a strategic approach. First, we must identify the unknowns and knowns in the problem. This step allows us to gain a clear understanding of what we aim to find and the information available to us.
Next, we must find an equation or set of equations that relate these unknowns and knowns. In certain situations, solving for one unknown may require the use of multiple equations, especially if the equation in question contains more than one unknown. One should always keep the relevant physical principles or mathematical properties in mind to ensure that the process does not deviate from logical reasoning.
In the case of an equation involving complex numbers, we pay special attention to the discriminant of the equation if it's a quadratic equation, as it indicates whether the solutions are real or complex. Once we substitute the known values into the equation, we should check to make sure that the solution is reasonable, including verifying the units and magnitude of any quantity with units.