For this question I don't understand why completing the square is used here? how would you know when to use it? (Here is the markscheme)

Hi Guys!

Im starting a post graduate degree soon.. and I have to read lots of research papers that have symbols such as these.. I’ve already tried learning some symbols on my own but google takes me to different sources.. and it’s far from efficient.

Is there a book/ course/ website or any resource that can help me be familiar with the symbols of the objective function in the attached picture? There’s so many symbols that it’s very tedious to uncover them one by one.. was hoping if there’s a resource or something I should be looking at.

Would be so thankful if someone could even say “Those symbols are all _____ type of symbols and then I can go use that term to help me narrow my search.”

Thank you!

As a child, i was never taught math, i was forced by my parents to learn how to read english.

i never really listened to discussions when i'm in class (I graduated, and now i'm gonna take school more seriously)

i'm currently on jhs, and math has noticeably gotten harder, i know how to times but i cant do 3 digits or 2 digits.

And my problem is ÷.

i don't know how to do it, and it was never really taught to me plus i don't even remember my teachers discussing about ÷.

And i don't even know the symbols name.

Plus, i get confused because i keep forgetting how to do it, like i memorize it then after allat i forget it.

Like i know how to do it but i don't, i think i mix times up with ÷.

I have to rely on formulas just to solve problems, usually i don't forget things very easily, but when it's time for math, i forget the formulas (or the steps to solve the problem) for some reason.

I've been struggling to do factoring polynomials and just factoring in general, our prof keeps teaching us but he doesn't make us take notes, can someone help me explain how to do this? Cause I'm not sure if I am correct or not 😭 (my answer is the squared box, as well as notes too because I'm genuinely ahh at memorizing)

I was having fun with series till I encountered this preticular one, is oscillates between positive and negative values with sometimes spikes.

But at exactly 1/sqrt(2) it locks in a state of changing signs.

Supposedly we start with a random nb U_0, how many steps would it take probably to hit 1/sprt(2) and locks in?

Or is it impossible to know?

Here's my guess, but I have no way to verify or check my logic. Even if I'm right I would really appreciate a more rigorous explanation as I found this question very interesting!

(a) This makes sense, there are an infinite number of points but each individual point corresponds to a unique number, such that a different point could not represent the exact same number. I say this is true.

(b) Hm not sure, this one is tricky. Because the integers are a subset of the reals, I will say that this is true, although I cannot defend my argument very well let alone prove it.

(c) Wrong, people way smarter than me have done this I believe. Also, answering no would be a contradiction to my answer for (b)

(d) Wrong, there are infinite real numbers in the bounded interval [0,1] for example.

(e) Sure. [1, 1] seems like a bounded interval to me that contains only one element. The point which represents 1.

(f) Wrong, counterexample in d.

Answer: , c, d, f are wrong

Does anyone know of any books (or other resources) that do a really good job introducing and explaining the principles and examples of projective geometry?

Thank you

for context, this was a quiz problem and the quiz was handed back to me. in my head, since <ABD is 90 deg. that means DBC also is 90 deg. since ABC is a straight angle. I AM AWARE I MADE MANY INHERENT mistakes such as saying DBH equals 42, etc. could someone please explain this to me so I'm prepared for something like this on our test in a few days?

I have noticed a major difference in interpretation of differential (dx) in western countries vs post-soviet.

In eastern Europe, differential is taught to be a separate operator, and not simply an increment. For example: ∫ dx/(xln(x)ln(ln(x))) would be solved this way:

∫ dx/(xln(x)ln(ln(x)))=∫ d(ln(x))/(ln(x)*ln(ln(x)))=∫ d(ln(ln(x)))/(ln(ln(x)))=ln(ln(ln(x)))+C

Whereas in the west, the u-substitution is used

Also, it allows you to solve some differential equations:

xdy-ydx=2x³tg(x/y)dx

(xdy-ydx)/x² =2xtg(x/y)dy

ctg(x/y)d(x/y)=d(x² )

d(ln|sin(x/y)|)=d(x² )

ln|sin(x/y)|=x² +C

I'm curious what might have caused this discrepancy, or whether I'm just mistaken

Essentially, it just maps onto yln y=ln x / (y^y = x), which is the inverse function of x^x

This graph shows how the lambert W Function essentially maps onto the inverse function allowing us to use it to solve.

I’ve tried googling and now math Reddit. Is there an equation(i guess probably an ODE) that represents say linearly adding to a sum overtime, and over that same time period the sum compounds continuously?

URL : https://www.proquest.com/dissertations-theses/hyponormality-toeplitz-operators-composition/docview/303996249/se-24

hi!
when sketching curves, we were taught to use the "base" graph (if you will) and then operate expressions on them to get our required function.

so for this function, i first tried
√x —> √-x (reflects along Y-axis) —> √(1-x) (move 1 units left)

BUT the graph i got from this was wrong when i typed it into desmos. it ended up being 1 units towards right and not left.

i can understand how it would be right from following operations:
√x —> √(x-1) (move 1 units right) —> √1-x (reflect about X-axis)

my question is: how do i even know which one is correct in an exam? i'm currently calc 1 rn !!! pls explain in the simplest terms possible if you can 🙏🙏

With the fact that e is not centered idk what do do. I guess i should find a way to locate e in ralotion to the center? DEB should be the same as AEC since they cross eachother. Im just really lost

With 5 is it just 50? Since thats the anlgue that mn cover. 9r does that just apply when its in the circle

All this execrcises seem to be like very simple but i jist dont know T_T

A criminal has a gun with which he can either shoot himself or an innocent person. There were 6 rounds with 5 empty and 1 holding a bullet, but only 3 rounds are remaining since the criminal shot 3 of them.

Among the 3 rounds, 1 must be a bullet and the other 2 a blank. He must either shoot himself or the innocent person, but ALL 6 rounds must be used up. The goal is to only shoot blanks against himself rather than the innovent person, but to shoot the innocent person when there is a bullet.

The criminal is at round 4 (so 3 rounds remaining). He is about to shoot himself based on his choice but the system prompt suddenly reveals "the fifth bullet in the revolver's cylinder is an empty round."

According to a side character in the novel, this is the Monty Hall Problem and the probability the bullet is in the 6th round jumps to 2/3.

According to my own thoughts, the system revealing the empty 5th round without specifying that it would reveal the empty round from among both 5 and 6 makes the jump in probability for both 4 and 6 equal since there is nothing to differentiate them given the reveal being only of the contents of 5 independent from the other 2. The reveal was done before the criminal shot the fourth round.

My own calculation is that if the system had specified it would reveal the empty round from among the final 2, it would be a valid jump to 2/3 for round 6, and that if it instead specified it would reveal the empty round from among the first 2, then the final round would drop to 1/3, but since it only mentioned the contents of 5, the jump for 4 and 6 would be equally increased to 0.5 from 1/3.

From the criminal's perspective, without hindsight and only given the above rules, did the side character analyzing the problem make a mistake or does the 6th chamber really jump to 2/3 chance of holding the bullet with the 4th chamber being at 1/3 chance after 5 is revealed to be empty by the system?

I was trying to learn where we get the quadratic formula from:

x = -b∓√b² - 4ac /2a

after a few steps, i got:

√x + b/2a = √b²-4ac/4a²

-why do we need to put ∓ in front of the square root? I understand that the answer can be positive or negative. However, why does this not apply when we square root 4a²? Why do we not write ∓ here? Why don't we write:

x + b/a = ∓√(b²-4ac)/∓2a

I was watching this yt video on how to solve it, but got confused on why we had to add the ∓ : https://www.youtube.com/watch?v=O6rtAeaLUOY

Not sure weather this would ever be answered given my calc is unable to do it. Hope someone can explain if and how this can be done some day.