Wtf it doesn't make sense, at all.
This is what I've come up with.
x^4-5x^2+4=0
(x^2-1)(x^2-4)
(x-1)(x+1)(x-2)(x-2)
x=-2,-1,1,2
I think it's right, but I've got no idea why I got where I did and none of the steps seem connected.
In an earlier post, I recollect that you said you memorized formulas rather than trying to understand principles. In that case, nothing will seem to make human sense because
you are acting like a machine rather than a human.
The principal involved in solving an equation in x is to change an obscure equation in x into an equivalent equation where it is obvious what number is represented by x. That is the simple principle involved. It is called "reductionism": we reduce the complex into simpler and simpler forms.
A lot of algebra is simply learning an inventory of techniques to reduce the complexity of an expression.
Just looking at
\(\displaystyle 4w^4 = 136w^2 - 900\) does not provide a solution that is immediately obvious to inspection.
One technique that you can try (it may not always be practical) is to set a polynomial equal to zero, factor the polynomial, and use the zero-product property.
\(\displaystyle 4w^4 = 136w^2 - 900 \implies 4w^4 - (136w^2 - 900) = 136w^2 - 900 - (136w^2 - 900) \implies 4w^4 - 136w^2 + 900 = 0.\)
The simple principles involved are that
\(\displaystyle a = b \iff a - c = b - c\) and \(\displaystyle d - d = 0.\)
Do you have any questions about the sense of those principles?
The purpose, however, is to transform the equation into a form that is a polynomial equal to zero because sometimes we can factor it to get a solution.
\(\displaystyle 4w^4 - 136w^2 + 900 = 0 \implies \dfrac{1}{4} * (4w^4 - 136w^2 + 900) = \dfrac{1}{4} * 0 \implies w^4 - 34w^2 + 225 = 0.\)
The simple principles involved are that
\(\displaystyle a = b \implies c * a = c * b \) and \(\displaystyle d * 0 = 0.\)
Have any wtf feelings about those principals?
The purpose is to simplify a polynomial into a somewhat simpler one. This is reductionism in practice: we try to simplify step by step.
We ask whether we can factor this polynomial: we can.
\(\displaystyle (w^2 - 25)(w^2 - 9) = (w^2 - 25) * w^2 - (w^2 - 25) * 9 = w^4 - 25w^2 - 9w^2 + 220 = w^4 - 34w^2 + 225.\)
\(\displaystyle \therefore w^4 - 34w^2 + 225 = 0 \implies (w^2 - 25)(w^2 - 9) = 0.\)
The simple principles involved are
\(\displaystyle a(b + c) \equiv ab + ac\) and \(\displaystyle a = b \text { and } b = c \implies a = c.\)
Do you find the principles obscure?
The purpose is to factor the polynomial. Why? Because that is one way to find an answer. How did we find the factoring? That is an art, not a science. So now we have
\(\displaystyle (w^2 - 25)(w^2 - 9) = 0.\)
There is a simple principle called the zero product property:
\(\displaystyle a * b = 0 \implies a = 0 \text { or } b = 0 \text { or } a = 0 = b.\)
Applying that we
\(\displaystyle (w^2 - 25)(w^2 - 9) = 0 \implies w^2 - 25 = 0 \text { or } w^2 - 9 = 0 \implies w^2 = 25 \text { or } w^2 = 9.\)
You can solve those by inspection. If you put those answers back into the original equation, you will find they work.
Our overall purpose was to transform an equation without an obvious answer into an equivalent form with an obvious answer. Our method was to try one of the techniques for doing that by applying gradual steps based on very simple principles.
If you don't understand what your destination is or why you want that destination, the directions to that destination will seem arbitrary. There is no sense to math by formula and rote; it is simply a set of magical incantations.
