$x^2+6x+8=0$
you can factorise this to $(x+4) \cdot (x+2)=0$ and it's quite obvious that there are two solutions and why they work. you can also solve it like this:
$x^2+6x+8=0$
$(x+3)^2+8-9=0$
$(x+3)^2=1$
$x+3=±1$
$x=-3±1$
now i understand every step, but why taking the square root gives the same two solutions feels kinda magical to me. it's even less intuetiv for me when there aren't any exponents in the first place.
1) $\tan x = 2$, find $\cos x$
2) $\tan x = 2$
3) $\frac{\sin x}{\cos x} = 2$
4) $\sin x = 2\cos x$
5) $\sin^2 x = 4\cos^2 x$
6) $\sin^2 x - 4\cos^2 x = 0$
7) $\sin^2 x + \cos^2 x = 5\cos^2 x$
8) $1 = 5\cos^2 x$
9) $\frac{1}{5} = \cos^2 x$
10) ±$\frac{1}{\sqrt{5}} = \cos x$
again, i know why this problem should have two solution, because i know the unit circle, but that taking the square root can find that other solution, i dont get at all. especially since the original expression didnt even have powers of two before i introduced them in step 5.
in other cases, if you square a number you can make a solution less accurate.
$x=-1$
$x^2=1$
$x=±1$
and then there was an instance where i dont have an example. i just remember a algebra problem where the teacher said that we shouldn't divide by sinx, because if we did that we would loose one of the two solutions, and i think i understood it in that particular instance, but i had no idea how i could see it with out being told, and what the general rule was.
so i have a few question.
what method are there to find more solution to problems? (e.g taking the square root)
in what ways can we loose solutions? (e.g squaring numbers, or dividing by certain numbers)
how do we know we have found all the solutions? (to e.g more difficult problems than those i mentioned)
also, is there an intuitive link between those examples i mentioned? like how factorising and taking the square root could lead you to the same two answers, even tough the methods semms different to me. and how taking the square root also is linked with the unit circle, even tough the expression "tan x" or "sinx/cosx" wasnt squared in the first place.