dsgrue3 said:
x^2 = 4. You're saying that -2 is not a solution? Just trying to understand where you're coming from. |
You're making a bit of a false dichotomy. I'm saying that "x = ±2" does not mean "both -2 and 2 are solutions". If I say that "J'ai faim" does not "I'm thirsty" in French, it doesn't mean I disagree with the result, only that it is not the correct interpretation.
The "both -2 and 2 are solutions" interpretation is *inclusive*, in that it actively claims that -2 works. It is a common misconception that when you apply algebraic manipulations to solve an equation, you are creating solutions from the equations at hand. In fact it is better to think of it as a winnowing process: initially x could be any real number, and you cannot hope to check all of them one at a time. The equations are clues which one actively uses to narrow down the possibilities to a manageable few.
[Many students "learn" to just throw out solutions "that don't make sense" without any understanding of the process. Those "extra" solutions are not really created from thin air: they were already among the infinite number of possibilities at the start, and the method simply failed to rule them out at an earlier stage. Sometimes the manipulations are fully reversible and there is a perfect correspondence between the solutions you find and the actual solutions. But this tends to happen only in trivially simple cases, and since these tend to be the first cases a learner sees, this misconception is easy to pick up.]
What I am saying is that the correct interpretation is *exclusive*, in other words "no number other than 2 or -2 is a solution". This makes no claim that -2 is a solution, only that -1 and 3 and uncountably many more numbers cannot be equal to x. But neither does it claim that -2 is NOT a solution. Starting from the assumption "x=2" and deducing that "x=±2" is not a contradiction (starting from "x=±2" and deducing that "x=2" would be flawed logic, but that is another discussion entirely).
