Substitution problem

Question

My question is something I've been thinking about for some time now. Q: Why is it possible to make substitutions or change in variables ? I mean, how do I know which substitutions are allowed ?

For example when we use Vieta's formulas to vanish with a second degree monomial. Or when we change variables to solve an integral equation.

Here is what I think:

if we want to solve an equation like this $ax^2+bx+c=0$ we could turn it into an equation like this one $Ay^2+B=0$ which we know how to solve, to do that we write $x=u+v$ and we get $a(u+v)^2+b(u+v)+c=0$ which is equivalent to $av^2 + (2au+b)v + au^2+bu+c=0$ that we know how to solve if we get rid of the term $(2au+b)v$ but to do that we make $u=-b/2a$.

But why are we allowed to write $u=-b/2a$ ? I think that we can do that because we are solving this equation in the set $\mathbb{R}$, and we can write any real number in the form $-b/2a$. That's why we can't solve $x^2+1=0$ by setting $x=\sqrt{y}$, in other words, because the function $\sqrt{}$ is not surjective. Finally, I think that we are allowed to make a substitution if the set we are trying to solve the equation in (in this case $\mathbb{R}$) allows us to write the number as the substution we make. Is that somewhat right?

Answer 1

You're a little confused but have the right ideas.

In the case of the quadratic formula, you write $x=u+v$ which is a redundant description of $x$ until you choose one of $u,v$ - there are infinitely many pairs $u,v$ giving some fixed $x$. Hence when we find it convenient for $u$ to be suddenly something specific, we can, because there is still $v$ left over. We definitely can't write any $u$ as $-b/2a$ here because $a,b$ are fixed, given numbers.

But what we can do is write any $x$ as $-b/2a + v$ where $a,b$ are fixed and $v$ is variable. The fact that there is exactly one $v$ for each $x$, or that $x\mapsto v$ is bijective is what makes this okay.

This is slightly stronger than your surjective condition simply to avoid annoying situations where you get several $v$s for one $x$ and miscount things as a result. (Important in e.g. integration changes of variable.) You could still do the substitution happily in this case, like saying $v^2=x$ when you know $x>0$. There are two possible $v$s here. It's just a bit annoying sometimes. The key thing here is just "any number $x$ can be written this way for some $v$" like you said.