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For context: I have a test tomorrow on algebraic expansion, factoring, and Pascals Triangle and something that commonly happens to me is that I usually omit coefficients or just forget exponents when doing my work. As an example, if I had 4(a^3)(b) when doing a binomial to the 4th, I either forget to cube the value “a” or multiply by 4. The test I have is relatively face paced (20 questions in 50 minutes) and there’s not much room to write out the complete expansion (we can’t use scrap paper). Any tips?

Edit: Images: https://imgur.com/a/DHrudBU

Looks like it doesn’t let me put images here but I can put them below. The main thing that was bothering me is that how do we know the remainder will be either 0 or a first degree polynomial. It seems unusual to see the remainder be a line because we usually just see it as a constant. The other point of confusion is that I’m pretty sure when you find the remainder then you express the answer as the quotient + (Remainder/divisor). So then how can it be true that p(x) = q(x) d(x) + r(x)? It should instead be p (x) = q(x) d (x) + ((r(x)/q(x)).

What are some exercises I can do to get better and faster at the Trachtenberg Method? I keep mentally mixing up numbers or fully forgetting the first number I had once I solve other parts. Any advice is welcomed!

I need help with question 10. I can't seem to figure it out. All the answers I come up with don't match the choices for what I can pick for the answers. What am I doing wrong? Which answer do you guys get, and please say how you got there in detail?

• Assuming the Popcorn bucket thing is cone shaped then
  • Radius = 3 in
  • Height = 5 in

• Use the cone-volume formula: V=13πr2h I keep getting 47.1 as my answer. If so the then the closest choice is 49 in. But unless I'm wrong the answer brightly is giving is 41.7 inch, is this a typo/glitch and the numbers have been transposed? like instead of 47.1 it's showing 41. 7? or have i just got the wrong idea? Should I have rounded up somewhere?

  https://brighterly.com/wp-content/uploads/2024/11/4-Math-test-for-grade-8-1024x683.png

Hi everyone.
Taking Alg. I as a college freshman again. I'm a readmit student and always struggled with math. Currently doing homework and have 20+ attempts on the below problem:

'Use the commutative property of multiplication to write an equivalent equation:

34 + 5m'

Can't use an equal sign within the equation as it kicks back an error on the website. I've tried writing it as:

34m + 5

(34)m + 5

5(m) + 34

5m x 34

Keep getting it marked wrong. If anyone could help me better understand what I'm doing wrong I would appreciate it greatly.

I have to show that ⋃P(x)=x for a discrete mathematics assignment, but I can't figure out what the symbol at the beginning means in this context. Its a Union, right? But what is is Union-ing with? It can't be as easy as I think it is, can it?

https://imgur.com/a/ndFgXxU

(Idk if this is the right place. I tried posting to askEngineering but I don’t have enough posts to participate. And on r/cars i think people would be to harsh if I'm wrong, i'm fragile)

When looking into the car’s range statistics to know if battery is an issue for the Nurburgring (Nürburgring is 20.832km) (I’ve done a lot of research on the road and I'm 100% certain the skirt will function). And initially I was dismayed; according to this supposed “accurate” brochure from McMurtry automotive themselves in a Schmee150 video (later I will use a number from the website itself). The range in f1 mode at Silverstone GP would just be 11.782km with voltage sag. That’s terrible. And I had already done arc calculations for every turn at the Nurburgring assuming it had the range in f1 mode. Getting a time of 5:09.69 (3.5G) to 5:14.05 (3G) depending if 3.5G is the new baseline maximum at low speeds or not.

So I thought “I need a real world comparison”. So I got the volkswagen idr to use its statistics to find the true range numbers. (And then the Rimac Nevera later to compare) The Volkswagen IDR had a 45kWh battery and used 24.7kWH (of which 9.7% was energy regenerated) for a time of 6:05.33 minutes. This is 54% of the entire battery used with all that downforce drag included. Which is 1.185kWh/km.

So to get a number comparison I would imagine it being at full power for the whole lap. So seeing as the idr used 54%. Ill multiply the full power result by 52% because of lower drag, and ill use the same time for starters.

So putting the Murtry into LMP2 mode: 650hp with 75% downforce (45kw fan). The fan energy has been calculated at a maximum of 60kW. So 650 x 745.7= 484.7kW  |   484.7 x 0.101389hours (which is 6:05.00, yes its 33 milliseconds slower in reality but for some reason i did that but the difference is so minor so whatever) which equals to 49.14kWh |  (49.14 x 0.52)= 25.55kWh | 25.55 x 9.7% =2.47kWh | 25.55 - 2.47= 23.08kWh | 23.08+4.56(fan)= 27.64kWh in reality | 1.326kWh/km

(I decided to add the fan at the end because the energy regenerated for the idr wasn’t with a fan included. Now the Murtry has crazy braking performance, 4G maximum with spoiler (according to info available currently)down to 3G constant baseline across all speeds and regenerates up to 200kW at maximum. I can’t calculate what energy it will regenerate since idk the exact numbers it will be going at before entering the turns, but I presume it would regenerate slightly less energy than 9.7% because of its braking performance and higher corner speed, but this is accurate enough and is satisfactory.)

 1.185kWh/km. IDR
 1.326kWh/km McMurtry

This is 12% more energy used which is reasonable. But now we’ll calculate using the brochure and the website for LMP2 mode.

From my interpretation, the brochure says 6.5 laps at lmp2 mode. Which is 100kWh in 38.291km. Which is 2.611kWh/km. The website displays a maximum of 50km in lmp2 mode, so 100kWh in 50km is 2kWh/km.

This is 168% to 220% times more energy used for a 35hp difference (plus a small amount of fan) on a car with a much smaller profile and drag. This seems unreasonably inefficient. The McMurtry is at the pinnacle of electric vehicle technology.

Now we’ll go to f1 mode: 1000hp and 100% downforce (60kW fan). We’ll use the 919 Evo’s time just to use a real number 5:19.54 minutes (0.08876 hours). The energy recovered would be less in reality I believe but i’ll use the same number, just imagine it being slightly bigger:  1000 x 745.7= 745.7kW  |  745.7 x 0.08876= 66.18kWh | 66.18 x 0.52 = 34.413kWh | 34.413 x 0.097= 3.338kWh | 34.413-3.338+5.32=  36.39kWh in reality | 1.747kWh

Now we’ll use the dreaded brochure: 100kWh in 11.782km = 8.487kWh/km | 8.487 x 20.832 (Nurburgring length) = 176.801kWh in 20.832km. Now depending how you interpret the brochure, the 52% could already be in there, but ill add the 52% anyways. [176.801kWh  &  8.487] times 52% is 91.936kWh | 4.413kWh/km !!!! To give a comparison now, let's see the Rimac Nevera.

The Rimac Nevera:

• 1914 hp
• 5071 lbs
• 120kWh battery pack
• Used 42kWh at the Nürburgring (7:05.298)
• 2.016kWh/km
• 10.39 MPGe

The McMurtry Spéirling Pure:

• 1000 hp
• 2971 lbs
• 100kWh battery pack
• 176.81kWh??? That’s 4.209x the Nevera in energy usage.
• 8.487kWh/km
• 2.46 MPGe

OR

• 91.936kWh??? That’s still 2.18x the Nevera.
• 4.413kWh/km
• 4.74 MPGe

Even if we interpret the brochure as the 60kWh battery pack, its still comes out at 5.093kWh/km or 4.11 MPGe. The reality at f1 mode should be 36.395kWh  -1.747kWh/km - 11.69MPGe for 5:19.34

The Pure has 914 less horsepower, weighs a whole 2094.76 pounds less. And completes the lap (according to my work) 115.608 SECONDS FASTER than the Rimac Nevera. Because the fan system allows it to do turns at incredible speeds with zero drag, and because it weighs 2094.76 pounds less, it requires much less energy in acceleration overall because its exit speed is much higher and it’s way lighter.

Referring back to the picture of the graph you can see this difference. Am I doing something wrong? Am I crazy to think this is impossibly inefficient for the numbers McMurtry Automotive is giving us? If i’m wrong, where is all this energy going? Because the fan barely draws power. The McMurtry is at the pinnacle of battery technology, motor technology, and the thermal load is lower. This shouldn’t be possible. All I can see is my numbers are the truth. If they’re like 95% accurate, why lie of range statistics this badly? This hurts the brand's image as it’s one of the main things haters say about the car is the range. But from this, its range is incredible for the performance. Are buyers brought in on this secret? Just like the road legal variant they’ve made zero mentions of which also hurts the image of the brand when not mentioning it (for 200kgpb you can make it road legal through Lazante, a photo exists reading the words “road conversion prototype” on it at GoodWood when the pure was on display). 

So what do you think? Am I validated in my logic?

My son just started High School algebra in 8th grade. His teacher gave him homework on which he said each equation needs a 1 step solution. My son asked me for help on the equation in the attached picture, I helped him get the answer but in 4 steps and I cannot figure out for the life of me how to simplify this to 1 step. Please help:

5r-5-6r=-6
Adding 5 to both sides becomes 5r-6r=-1
-r=-1
r=1

here’s a picture of the problem

(TLDR at the bottom of my post) hello everyone!

i havent had math since i was 16 (now 23) and all my life i have always been bad at it... or so i thought! i have to take up math at school again if i am to get into university to study biology, so ive been studying and learning everything from the bottom up all summer! i must say ive been having quite some fun, and it turns out im not as bad as ive been telling myself my whole life:) i am very proud of myself for trying and for keeping on going! ive learned so much in such a short period of time!

ive started school recently and we are currently dealing with logarithmic equations. i know all the rules of exponents, log rules, and quadratic equation by heart, but there are some things i still struggle with. i really do NOT want to use AI but i dont have access to my math teacher 24/7, and i want to understand things on my own. i want to know what the next step is, what the best approach for an equation is, and why it works. also knowing when i should apply what rule/logic/approach would be great, but im assuming it is mostly pattern recognition. i get most of it on my own even tho i make a lot of silly mistakes, but some of them i really dont understand what im supposed to do

i REALLY want to understand the process and why everything works, and why we do it this way. i want to understand my mistakes

here are three things i am struggling with this evening, number 3 being the one that confuses me the most. they are my attempts at solving each equations, so the examples in themselves might be wrong (i wonder if number 2 is wrong too)

ln(x+1) = 1

e^ln(x+1)=e¹ (-> e¹ = e)

x + 1 = e

x = e - 1

my question for this one is: why do we put e as the coefficient? why do we raise ln(x+1) and 1 to be the powers of e? and as a follow up question, why do we then get rid of "e^lnx" part? why are we then only left with x + 1 = e? i understand that e¹=e, but i dont understand why we raise everything and then get rid of a part of it. how do i know when to do it, and how do i know what i get to remove after i have raised everything up?

ln(x/5) = -1

lnx = -5 (-> we multiply both sides by 5 to get rid of the fraction)

x = e^-5 = 1/e^5

i know that lnx = a <-> x = e^a, but why cant i apply the log rule of lg(a/b) = lga - lgb? and then raise everything to the power of e, like we did in example 1? why not this?

ln(x/5) = -1

e^{lnx - ln5} = e^-1 = 1/e

x - 5 = 1/e

x = 1/e + 5

1. (and this is the one giving me the most trouble)

ln(x+3) * (lnx-3) = 0

for this one, i dont even know where to start:( my teacher said that here, either ln(x+3) or ln(x-3) have to be equal to 0, but i dont really know what that means. i feel like im so close to the answer but something is in the way! what is the next step to this equation?

im sorry if this is all confusing, i tried to make it as readable as possible!! thank you for any helpful information<33

TLDR:
what is the best approach to this equation? what is a step by step list to follow?

ln(x+3) * ln(x-3) = 0

I am a graduating senior in high school and one of my problems is that my math solving skills is so bad. I can't even solve a basic double-digit subtraction, and I am currently applying to a few prestige universities in my country, but the problem is I don't think I will be able to cram almost 17 years of math knowledge in just a month. I am very educated in other subjects needed for the exams, but I am extremely worried about the math proficiency exam. any advice?

in my class we have just finished functions and after we finish a topic it takes me really long to fully understand it.
here is the questions:

3a) write 2x ^2 -8x -3 in the form a(x+p) ^2 + q

b) Use your solution to find the turning point and roots. Also find the y-intercept then draw and sketch, clearly all these points.

i did part a) and got 2(x-2)^2 -11

and for the tp i got (-2,11) and y intercept i got (0,-3)

i forgot how to find the roots however. i think the shape is a parabola??

4) On the same diagram, make sketches of the following functions, showing the important points.

a) log4x
b) log4 (x-1)

—————————————————————————————
we actually haven’t done logs yet, but my maths teacher told us that it is the inverse function i think??

i’m really grateful for this community to help me 🙏. i used to use chatgpt for my maths hw, but i am really trying to understand the maths behind it so i can improve

Taking applied calculus and we are going over functions to begin stuff. I’ve done all sort of stuff with functions and graphing before and most of this has been pretty easy but there are two problems that have been given that I just can’t seem to figure out and I’m not sure what I’m doing wrong.

The first is:

“Find the function g whose graph is constructed from the graph of f by a shift to the right by 10 followed by a horizontal stretch away from the y-axis by a factor of 3?”

I thought it would be pretty easy but what I thought it was didn’t work, and every other possible combination hasn’t worked either so I’m completely lost. I’ve done (1/3x - 10) and (x/3 - 10) in all sorts of ways. I’ve removed the parentheses, I’ve added negatives, doing a whole bunch of shit that I know isn’t right but I can’t figure out what else to do.

The second problem is along the same lines:

“Find the function g whose graph is constructed from the graph of f by applying the following steps: shift to the right; horizontal stretch away from the y-axis by a factor of 4; a vertical shift up by 5; a vertical stretch away from the x-axis by a factor of 7.”

Again I’ve tried all combinations I could think of but I can’t seem to get it. I’m not sure if I’m missing something? I don’t believe my format to be off since it automatically starts with g(x) = , so that shouldn’t be the issue. The entire set of problems is all working on vertical and horizontal shifts, stretches, and compressions and I haven’t had an issue with any of the problems but these two. Please help me figure out what I’m missing!

Im currently a first year student studying comp sci and im taking discrete math the issue is unlike any other faculty i have to learn actual theory behind math like whats C(n,k) selection and explain it with words and formulas.Ive got about 12 pages filled with 50 or 60 questions which if memorized gives me a passing grade and ive got 2 days to do it keep in mind the professor wants the questions answered exactly by his pdf. Ive youve had the same issues how should i go about tackling the problem and passing matu

Hello, I am interested in finding a way to train myself with fractions or math calculation techniques in chemical and physics formulas. How do I do that?

This is for a medschool entrance exam

I'm preparing for a competitive exam and I'm struggling with math even though I understand most of the concepts and formulas.

For example, in trigonometry I know the basic identities, values, signs, radians, ranges, etc. I've also solved many questions. But when I get a mixed or unfamiliar question, I often don't know what path to take. Sometimes I know the approach but get completely stuck in the algebraic manipulation.

Even after practicing a lot, I still feel like I'm forgetting the process of solving.

What should I actually work on to improve this? Is it mainly algebraic manipulation, problem-solving skills, or something else?

Sorry guys I know I’m dumb af. But why isn’t this showing the answer as 2.5? I understand the fraction answer is correct but I just want it to say 2.5 😭 I tried different settings and can’t get it. https://imgur.com/a/JS6SHY9

The question is (6x+16)/4 + (6x-1)/4 but in fractions. I believe that it's (12x+15)/4 , but the answer sheet says that it is (12x+17)/4 instead. Google says that I am right. Am I right?

hello, i had dana fryer's math questions downloaded part 1 & 2 , and was wondering if anyone had the correct answers(from math bootcamp)? i do not have access to them anymore. :/

Im in class 12th from allen online enthuse, completed the whole syll but since having adhd I couldn;t take live class actively I hv all my basics and concept clear but there is the massive logic gap and execution gap, what should I do only 4 months remaining, I am panincking for chapters like integration, differential equation and matrices which require self talk and active solving, IDK the fuck anything about logic building...
for example_ applying bornouli eq in differential equation when eq having apecific form of dy/dx+py=qyn, and appying ibp to saperate the terms out of integral and reducing the powers... actually im in a problem tht these gaps are in large quantity

I'm taking algebra 2 and we're reviewing factoring right now. Since polynomials live in the real numbers, it's not clear to me why we limit ourselves to integers when factoring.

If I was factoring 6, why isn't 1.5 * 4 just as valid as 2 * 3? I realize factoring beyond integers would mean there are an infinite number of factors...but I'm not sure that alone is justification.

I've been at this for like 4 hours and I'm losing my mind.

So I'm thinking of a function that takes in four boolean inputs and outputs 1 if the number of inputs is odd. The truth table should look like:

A  B  C  D | OUT
0  0  0  0 |  0
0  0  0  1 |  1
0  0  1  0 |  1
0  0  1  1 |  0
0  1  0  0 |  1
0  1  0  1 |  0
0  1  1  0 |  0
0  1  1  1 |  1
1  0  0  0 |  1
1  0  0  1 |  0
1  0  1  0 |  0
1  0  1  1 |  1
1  1  0  0 |  0
1  1  0  1 |  1
1  1  1  0 |  1
1  1  1  1 |  0

So the function should just be the sum of products, right? It'd be something like

OUT = A'B'C'D + A'B'CD' + A'BC'D' + A'BCD + AB'C'D' + AB'CD + ABC'D + ABCD'

This is my attempt to simplify the expression:

A'B'C'D + A'B'CD' + A'BC'D' + A'BCD + AB'C'D' + AB'CD + ABC'D + ABCD'
= A'B'(C'D + CD') + A'B(C'D' + CD) + AB'(C'D' + CD) + AB(C'D + CD')
= (C'D + CD')(A'B' + AB) + (C'D' + CD)(A'B + AB')
= (C'D + CD')(1) + (1)(A'B + AB')
= A'B + AB' + C'D + CD'

But the truth table for this expression is not the same as the original function.

It's been a half-decade since I last did boolean algebra. I genuinely can't tell what I did wrong here. Please help.

Edit: It's solved. Thank you.

In the blindingly-unlikely event someone in the future were to face this exact problem as I do, the key insights that I was missing are:

• OR distributes over AND: A + BC = (A + B)(A + C)
• XOR and XNOR exists: A XOR B = A ⊕ B = A'B + AB', and A XNOR B = (A ⊕ B)' = A'B' + AB

So for posterity's sake, here's how it's solved, I think:

A'B'C'D + A'B'CD' + A'BC'D' + A'BCD + AB'C'D' + AB'CD + ABC'D + ABCD'  
= A'B'(C'D + CD') + A'B(C'D' + CD) + AB'(C'D' + CD) + AB(C'D + CD')  
= (C'D + CD')(A'B' + AB) + (C'D' + CD)(A'B + AB')  
= (C ⊕ D)(A ⊕ B)' + (C ⊕ D)'(A ⊕ B)  

Let P = A ⊕ B and Q = C ⊕ D  

= QP' + PQ'  
= P ⊕ Q  
= A ⊕ B ⊕ C ⊕ D  

Thanks, y'all. god, i'm out of practice.

okay so im working on the study guide for the final of a class that i took the equivalent to at another college (math department policy, whatever) and one of the sample questions literally just reads “simplify: (sin(x)+3)²”. is there anything that i need to do to work this problem other than factoring it out? a similar question in the same section of the review guide is “simplify: sin²x + 3sinx - 10”; is that not already fully simplified? like i said, i took this course at another school so i don’t know the exact details of what they covered in class. I can’t think of any identities that i could apply to simplify it more or anything. much appreciated!