Intuition behind Algebra When proving, or just straight up solving equations, we often manipulate variables until we get the results we want. For example, the square root of the discriminant is simply an algebraic manipulation to get the result of the distance (or difference) between the roots. However, is there intuition behind this manipulation? I get why Algebra works in the first place but when seeing it in action, manipulations as complex as that of involved in getting the discriminant, gets me lose sense of what's actually happening within the numbers.

An equation like $x+4 = 3x$ returns True for some values of $x$ and False for others.
For example, this particular equation returns false for $x := 1$, because
$$(x+4 = 3x)(x:=1) \iff 1+4= 3\cdot 1 \iff 5 =3 \iff \mathrm{False}.$$
Whereas it returns True for $x:=2$, because
$$(x+4 = 3x)(x:=2) \iff 2+4= 3\cdot 2 \iff 6 = 6 \iff \mathrm{True}.$$
In summary, each equation is associated with a function that returns True for some values of $x$ and False for others.
When you solve equations, you perform manipulations that preserve this function.
For example:


*

*The equation $x+3 = 5$ is associated to a function that returns True for some values of $x$ and false for others.

*The equation $x = 5-3$ is also associated to such a function.

*Furthermore, these functions are equal.
Therefore:


*

*if $x+3=5$ returns True for some choice of $x$, then so too does $x=5-3$, and vice versa.

*if $x+3=5$ returns False for some choice of $x$, then so too does $x=5-3$, and vice versa.
This may not sound too deep, but it allows us to find the exact values for which an equation is true. If I write $3x+6 = 0$, it's a bit mysterious which values of $x$ make this function return True. But I write $x=5$, it's really obvious. For instance:
$$(x = 5)(x:=5) \iff 5 = 5 \iff \mathrm{True}$$
$$(x = 5)(x:=4) \iff 4 = 5 \iff \mathrm{False}$$
Surprise surprise, the equation $x=5$ seems to return True at $x:=a$ if and only if $a$ equals $5$.
What this means is that if I can repeatedly transform $3x+6= 0$ into a form like $x=5$, then I can work out which choices of $x$ make it true really easily. This is what solving equations is all about.
For example, if I write $$3x+6 = 0 \iff 3x = -6 \iff x = -6/3 \iff x = -2,$$
this instantaneously tells me that $3x+6=0$ returns True for $x:=-2$ and False otherwise.
That's really all algebra is; it's about manipulating conditions so that the condition hasn't changed, just the way we've written it has changed. The way we've written the final expression may tell us something useful about the original expression which wasn't obvious until we performed the manipulations.
