What is the order of operations for solving any variable? I assumed it was the opposite of PEMDAS, but in $2x^2 = 4$, you have to deal with the exponent first before dividing out to get a correct answer which breaks the order. Unlike $5^{(2x^2-1)} = 25$ which when you make the bases the same is $...
PEMDAS isn't how we do it in math. Instead we use the axioms of a field. This avoids all the problems that come with PEMDAS. mathworld.wolfram.com/FieldAxioms.html
If there is only one occurence of $x$ in your equation, the "rule" is to "undress", layer by layer, the term containing $x$. For instance in $2x^2=4$, you begin with dividing both sides by $2$, to "isolate" $x^2$. Whereas in $2x^2-1=2$, you first add $1$ on both sides.
The biggest difference between the two equations is that in one of them: $x$ is in the base, in the other one, it is in the exponent. You always want to isolate the unknown. There is no universal order of solving, since it depends on where the unknown is. Also PEMDAS is about order of evaluating formulas when there might be confusion possible (which still works in fields, contrary to what cyclo said, it's just a way to remember the right order) and not about solving equations. You want to isolate $x$ using a layer-by-layer approach just like Anne mentionned.
I don't understand your first example, Haridasa. $2x^2=4$ is correctly interpreted as $2(x^2)=4$, not $(2x)^2=4$, and the first step I would take in solving for $x$ is to divide both sides by $2$, rather than to do anything involving the exponent.
You have to undress layer by layer. You cannot take off your socks before taking off your shoes. $2x^2=times(2,power(x,2))$. Here, times is the shoe and power is the sock. It is inside.
You are missing the point, Haridasa. When you write 2x^2=4, it appears that what you mean is $(2x)^2=4$, but everyone else in the world interprets it as $2(x^2)=4$, because that's what the accepted conventions of mathematical notation tell us to do. If you mean $(2x)^2=4$, then you have to write it as $(2x)^2=4$. Now PEDMAS says, to evaluate, first do the parenthetical $(2x)$, then exponentiate. To solve, first undo the exponentiation, then unravel the parenthesized term. Socks & shoes, as @Anne wrote.
Interesting, but then how do we know how to correctly interpret parentheses what are the rules/proofs for such? Is there a place where I can publicly find all modern math rules?
@Haridasa the rules are the axioms of a field and parenthesis always have priority. That allows you to distinguish between $2x^2$ and $(2x) ^2$. That's all the rules.
It seems, Haridasa, that you still don't understand that $2x^2$ means $(2)(x^2)$ and not $(2x)^2$, so when solving $2x^2=4$ it makes perfect sense to divide by two before dealing with the exponent.
Axioms have nothing to do with it, Haridasa, it's just PEDMAS. E-for-exponent takes precedence over M-for-multiplication, so in $2x^2$, first you do the exponentiation, $x^2$, then you do the multiplication, $(2)\times(x^2)$.
As you have been told, Haridasa, pemdas is a set of rules for evaluating terms, not for solving equations. Evaluating a term is putting on your socks before you put on your shoes. Solving an equation is taking off your shoes before taking off your socks. Evaluating and solving follow different – indeed, opposite – rules of precedence. You can't break pemdas when solving, because pemdas isn't made for solving, it's made for evaluating. You can't break the rules of chess by making an illegal bid at bridge.
I'm not sure what you mean by "deconstruct". I don't know whether anyone has ever written down the equivalent, for solving, of rules of precedence. I'm not even sure the concept applies, since you evaluate formulas but you solve equations. You evaluate $2x^2+3x-4$ using pedmas, but you solve $2x^2+3x-4=0$ by using the quadratic formula, not by undoing the building-up of the quadratic.
Discussion on question by Haridasa: O…
Discussion on question by Haridasa: O…