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Consider $x^2+3=0$. If we add $x$ to both sides to construct a new statement $x^2+x+3=x$, what type of reasoning / logic is this?

Another example would be completing the square. New students might say that the added term when completing the square "comes out of nowhere." Surely it can be done because (and forgive my elementary language) "whatever is done to one side can (and must) be done to the other," but what type of logic is the action of doing so?

Are there recommended ways of explaining this concept to K-12 math students?

Gage Sorrell
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  • Be careful, because it is not always true that "whatever is done to one side can (and must) be done to the other" in terms of solutions. For example, $y=x$ is a line. But you get a tautology if you multiply both sides by $0$: $0=0$, which is true no matter the values of x, y. – amWhy Oct 07 '20 at 21:34
  • Similarly, if you take the square root of each side of the equation $x=-3$, where $x$ has a real solution, to get $\sqrt x = \sqrt{-3}$, you've lost your real solution. – amWhy Oct 07 '20 at 21:38

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If two quantities are equal you can always replace one with the other.

If $a=b$, the quantities $a$ and $b$ are equal, then in the expression $a+x = a+x$ you can replace one of the $a$'s with $b$ to get $a+x = b+x$

jjagmath
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  • You address addition (and subtraction) only. You ruin the information in an equation if you multiply both sides by $0$. – amWhy Oct 07 '20 at 21:35
  • But what kind of reasoning are we using if we say that "if this algebraic statement is true, then this algebraic statement with something new added to it is true"? (Where "added" might not mean literal addition, but some new term thrown in somewhere). – Gage Sorrell Oct 07 '20 at 21:41
  • @GageSorrell What do you mean by "what kind of reasoning"? It's not clear what you're looking for. This is sometimes called the substitution law of equality or some such name. – Alex Kruckman Oct 07 '20 at 22:35