Let's take a guess and set and see where this assumption leads. If we want to claim we have actually found the two solutions, we have an extra couple of steps. We avoided rationalizing before the very last step, which is OK, but when radicals appear in the denominator of our answers, this must be fixed. A lazy person could very well argue after all this that the quadratic formula is quicker. It is true, but the value of the above process is practicing looking at a set of equations for clues, using those clues to get answers, and then checking the answers to see if they make sense.
Solution Let's drag in our coefficients again, as a memory and organizational aid.
The thing to do is choose a definite value for one of the coefficients, then go through the equations to get all the others consistent. So, let's choose the most obvious possible value for the first constant. We'll talk about it in the same context as the above examples, again because we're trying to show that the mechanics are pretty much the same for all these problems. Since we got our original quadratic back, we're done. This is a bit more systematic, but still involves guessing.