I'm taking Algebra again in college again next semester and wanted to ask about radical solutions.
Okay, go get an apron, a butcher's knife and some intel on the nearest hog lot. You do know where your prof lives, right?
In the meantime I'll try to answer your question:
I have an example in my book where it feels that the numbers simply come out of the blue for some reason and I don't know why they exist. "What do you mean?" you might be asking, well its this.
Imagine a radical stretching over (X-1) and it continues to "equal X-7". From here I understand to get rid of the radical you do the opposite which is exponents.
Which nets me X-1=X(to the second) - 14x +49. (Note: I don't know how to do exponents on my keyboard, I apologize in advance).
At this point I have to ask why 14x and 49 are existing simultaneously. I guess it has to do with the X being multiplied by the 2 but if thats true then there should only be 2 X's and not 3.
The problem is sqrt(x-1) = x - 7, right? Your approach is perfectly valid, it seems you're just having problems with squaring the x - 7 term.
Remember: x behaves just like a number most of the time. You just multiply like you'd multiply any sum:
(x - 7)^2
= (x - 7)*(x - 7)
= x*x - 7*x - x*7 + 49 (Is this bit clear? You really just dissolve the parentheses.)
= x^2 - 7*x - 7*x + 49 (Since x^2 is defined as x*x and multiplication is commutative
)
= x^2 - (7 + 7)*x + 49 (We reintroduce some parentheses to simplify)
= x^2 - 14*x + 49 Voila!
Couple of style tips: Use lower case letters for variables - uppercase ones are usually reserved for sets or probability distributions (or something like that, I don't quite remember). And (as you've seen above) roots of something are usually written as sqrt(something) when writing plain text.
A question of my own: On
p 410 of Hatcher's Algebraic Topology he says that
To the extent that fibrations can be regarded as twisted products, up to homotopy equivalence, the spaces X_n in a Postnikov tower for X can be thought of as twisted products of Eilenberg-MacLane spaces K(\pi_n(X),n).
What does he mean by 'twisted products'? Spaces that locally look like products but globally have a different structure? Then I'd need to know whether the twisted product of two finite CW complexes is still finite...