The way I'd explain this is the reverse of the way you're going about it. You're starting with a complicated equation and simplifying it. I'd start with the s

The way I'd explain this is the reverse of the way you're going about it. You're starting with a complicated equation and simplifying it. I'd start with the simplified equations and make them gradually more complicated, showing carefully at each stage how to get back to the simpler case. 0. We can solve linear equations. 1. If you have a factored quadratic polynomial equal to zero, you can find its solutions by solving F_i = 0 for each factor F_i (which we can do, by 0, since F_i is linear). A solution to either of those linear equations must be a solution to the original equation by zero product property. 2. If you have a quadratic expression equal to zero, you can find the solutions by factoring, then applying (1). 3. If you have an equation L = R with quadratic expressions on both sides, you can find its solution by subtracting R from both sides to get L-R = 0, then applying (2). Both rigorous and simple, and also introduces one of the standard design patterns of higher mathematics: Find a solution to a simpler problem, then see what kind of more-complicated problems you can solve by reduction to the simpler case.