Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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4
votes
4answers
369 views

Functions of algebra that deal with real number

If the function $f$ satisfies the equation $f(x+y)=f(x)+f(y)$ for every pair of real numbers $x$ and $y$, what are the possible values of $f(0)$? A.  Any real number B.  Any ...
0
votes
4answers
54 views

Algebra that includes functions and graphing

The answer to the following is B. Can someone explain me how it is please?
6
votes
2answers
159 views

Finding the coefficient

How to find the coefficient of $a^3b^4c^5$ in the expansion of $(ab+bc+ca)^6$
3
votes
1answer
44 views

$ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right).$

Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: $ \log_{\frac ...
5
votes
2answers
330 views

Can the distance from the vertices of a square of integer width to an inscribed circle all be integer?

I'm looking for solutions to the following British Mathematical Olympiad question: Suppose that $ABCD$ is a square and that $P$ is a point which is on the circle inscribed in the square. Determine ...
3
votes
4answers
78 views

$(\frac1{n^{\sqrt n}})^{\frac1n}=(n^{-\frac{\sqrt n}{n}})^{}$

I'm a little bit confused by this one. Is this correct? $$\left(\frac1{n^{\sqrt n}}\right)^{\frac1n}=\left(n^{-\frac{\sqrt n}{n}}\right)^{}=\sqrt{n^{-\frac1n}}$$ ${}{}{}{}{}{}{}{}{}{}{}{}$ Edit: Is ...
0
votes
1answer
51 views

Corroboration Of Simple Algebra For A Physics Lab

I have a few equations that I need to solve for a specific variable, and I am wondering if anyone would care to look them over. The first equation is deriving Kepler's equation of orbital period, and ...
0
votes
2answers
172 views

Proofs with real numbers and discrete mathematics! [duplicate]

Can anyone assist me in figuring out the way to tackle the question Let $x$ be a real number such that $x > 0$. Prove that $x + (9/x) ≥ 6$. I understand that its true because, for example, let $x ...
0
votes
3answers
90 views

Solving the following equation: $2^{x}+\log_{10}x-2=0$

How can I solve the following equation: $$2^{x}+\log_{10}x-2=0$$ Any help welcome. Thanks!
2
votes
3answers
1k views

The area of the triangle with vertices $(3, 2), (3, 8)$, and $(x, y)$ is $24$. What is $x$?

The area of the triangle with vertices $(3, 2), (3, 8)$, and $(x, y)$ is $24$. A possible value for $x$ is: a) $7$ b) $9$ c) $11$ d) $13$ e) $15$ Please show your work and explain.
0
votes
2answers
163 views

Engineering economy total cost

A firm operators in a perfectly competitive market whose total cost varies as $TC = X^3 - 3X^2 -10X + 2$, and the price of the product they manufacture is given by $P = 130 - 2X$, where $X$ is number ...
0
votes
2answers
103 views

Finding $\frac{1}{d_1}+\frac{1}{d_2}+\frac{1}{d_3}+…+\frac{1}{d_k}$

If we assume that $d_1,d_2,d_3,...,d_k$ are the divisors for the positive integer $n$ except $1,n$ if $d_1+d_2+d_3+...+ d_k=72$ then how to find ...
4
votes
4answers
219 views

Working with proofs help?

I'm trying to study for my midterm and doing some random practise questions to work with proofs. However I'm stuck on, as the only way I know how to prove it is through plugging in numbers, however as ...
2
votes
3answers
319 views

Find all pairs of positive whole numbers

Find all pairs of positive whole numbers x and y which are a solution for $ \dfrac{2}{x} + \dfrac {3}{y} = 1 $. I don't really understand how to tackle this question. I rewrote $ \dfrac{2}{x} + ...
2
votes
1answer
84 views

Multiplying square roots

How do I simplify the following types of question: $ \sqrt{x^2+5} \times \sqrt{x^2+20}$ Do I need to get both answer out of their roots first or not? This is how I would do it: $ \sqrt{x^2+5} ...
7
votes
4answers
223 views

How to solve equation $ \frac{1}{2} (\sqrt{x^2-16} + \sqrt{x^2-9}) = 1$?

$$ \dfrac{1}{2} (\sqrt{x^2-16} + \sqrt{x^2-9}) = 1$$ How can I solve this equation in the easiest way?
2
votes
6answers
132 views

The remainder of $1^2+3^2+5^2+7^2+\cdots+1013^2$ divided by $8$

How to find the remainder of $1^2+3^2+5^2+7^2+\cdots+1013^2$ divided by $8$
2
votes
1answer
77 views

Proving $\sum \limits_{i=1}^k n_i^2 \le n^2 -(k-1)(2n-k) $

Given, $\sum \limits_{i=1}^k n_i = n$ and $n_i \ge 1$. Prove that $$\sum \limits_{i=1}^k n_i^2 \le n^2 -(k-1)(2n-k) $$ I am facing some problems in understanding the following step of this proof: $$ ...
0
votes
2answers
30 views

Solve Parameters of a Limit

$$\lim_{x \rightarrow 1} \frac{\sqrt{a-x} - \sqrt{b+x}}{x-1} = -\frac 1 2$$ What is $a$ and $b$? I just need a general direction to solve these type of questions. Thanks
2
votes
3answers
217 views

Can $n(n+1)2^{n-2} = \sum_{i=1}^{n} i^2 \binom{n}{i}$ be derived from the binomial theorem?

Can this identity be derived from the binomial theorem? $$n(n+1)2^{n-2} = \sum_{i=1}^{n} i^2 \binom{n}{i}$$ I tried starting from $2^n = \displaystyle\sum_{i=0}^{n} \binom{n}{i}$ and dividing it ...
0
votes
1answer
37 views

Plug $2m≥5f$ into $m-n+f=2$ and simplify

Plug $2m≥5f$ into $m-n+f=2$ and simplify It is a question on Euler's formula but I can't seem to simplify it when I try to add the inequality to Euler's formula.
1
vote
1answer
74 views

Problems with basic algebra

I'm studying for an exam in a digital communications course I'm taking, and the solution to one question has me totally lost. While finding the Inverse Fourier Transform of a function, there's one ...
3
votes
2answers
181 views

Solve $x = \frac{1}{2}\tan(x)$

I did this using trial and error, but I was just wandering if there is an algebraic way of solving this? I thought about double angle formula but that doesn't work properly does it? I then tried ...
0
votes
1answer
53 views

Solve $(1+1/x)^x=(1+1/a)^a$.

Solve $(1+1/x)^x=(1+1/a)^a$. Obviously, x = a is a solution. How can I know are there any other solutions? Thanks.
1
vote
1answer
172 views

perimeter and side length linear equation section

The perimeter of a regular hexagon is 3.04cm less than the perimeter of a regular hexagon. The perimeter of the regular hexagon is 21.06. What is the side length of the regular pentagon?
1
vote
5answers
321 views

How to prove that $\frac{x}{a} + \frac{y}{b} = 1$ where $a$ is $x$-intercept and $b$ is $y$-intercept

How to prove that $\dfrac{x}{a} + \dfrac{y}{b} = 1\;$ where $\,a\,$ is the $\,x$-intercept and $\,b\,$ is the $\,y$-intercept for all $\,a,b \neq 0$ This was a question on my son's math analysis ...
0
votes
1answer
38 views

relation between the coordinates of the vector on the unit sphere

Let $x=(x_1, \ldots, x_n)$ be a vector on $S^{n-1}$. Reorder coordinates such that $|x_1|\leq \ldots \leq |x_n|$. I am wondering if there is a some relation between the absolute value of the ...
1
vote
2answers
199 views

Indices - Numbers as a product of prime numbers

I've checked the internet which only provides basic $x^2 \times x^3 = x^5$ information and have concluded that I need resort to a Q & A website. The basics of indices are fine for me, but it's ...
0
votes
0answers
61 views

logistic difference equation problem

Consider logistic difference equation $${{x}_{n+1}}-r{{x}_{n}}\left( 1-{{x}_{n}} \right)=f\left( x \right),\ \ 0\le {{x}_{n}}\le 1\ \ \ \ \ \ \left( 1 \right)$$ 1.Show hat expression $$f\left( f\left( ...
2
votes
5answers
245 views

Prove the identity $1 + \sin x = 2 \cos^2 \left(45° - \frac{x}{2}\right)$

Here is the problem: $$1 + \sin x = 2 \cos^2 \left(45° - \frac{x}{2}\right)$$ Can you help me prove that this is an trigonometric identity?
1
vote
1answer
135 views

Another trigonometric equation

Show that : $$31+8\sqrt{15}=16(1+\cos 6^{\circ})(1+\cos 42^{\circ})(1+\cos 66^{\circ})(1-\cos 78^{\circ})$$
1
vote
0answers
131 views

Some trigonometric equation problems

show that : $$\left(1+\cos \frac{2\pi}{13}\right)\left(1-\cos \frac{4\pi}{13}\right)\left(1+\cos \frac{6\pi}{13}\right)\left(1+\cos \frac{8\pi}{13}\right)\left(1-\cos ...
1
vote
1answer
47 views

how to find cosinus betwen two vector?

i have task in linear-algebra. Condition: we have triangle angles A(-4,2); B(-1,6); C(8,-3); How to find cosinus between BA and BC vectors? please help :( what ...
9
votes
4answers
704 views

Proof of inequality $e^x + e^{-x} \leq 2e^{x^2}$

How would I prove that $$e^x + e^{-x} \leq 2e^{x^2}, \quad \text{for all real $x$}?$$ I narrowed it down to proving for $x \in (-1,1)$. I observed that for $(0,1)$ and for $(-1,0)$ I may need to ...
20
votes
4answers
520 views

Evaluating $\sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + \ldots}}}}$

Inspired by Ramanujan's problem and solution of $\sqrt{1 + 2\sqrt{1 + 3\sqrt{1 + \ldots}}}$, I decided to attempt evaluating the infinite radical $$ \sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + \ldots}}}} ...
0
votes
1answer
713 views

Height and width of x rectangles to fit in area

I have an area of $A = 320 \times 480$. I then have a number of rectangles of max size $128 \times 152$ which need to fit into the area. There can either be $1$ rectangle or $100$ rectangles. How ...
1
vote
4answers
674 views

Solving equations with high level exponents

Well I am doing my homework, and I can't seem to figure out how to solve this one: Find all solutions, real and complex, of the equation. (Enter your answers as a comma-separated list. If there is ...
0
votes
2answers
73 views

I have a really dumb algebra question for you

Suppose $k_1, k_2, k_3 > 0$. Suppose $x \leq -k_3$ and $y \leq k_2$. What, if anything, can you say about $\frac{x}{y}$? What if $y \geq k_2$? Also, what can you say about $\frac{y}{x}$? ...
0
votes
1answer
33 views

Simplification changes the output?

if I have a function $ f(x)=\frac{2x}{x+1}$ I can simplify it to $f(x) =\frac{2}{1}=2$ But this changes the function for example $ f(10)=\frac{20}{11} $ which is not equal to 2. Does this mean if I ...
3
votes
3answers
149 views

Please help me to solve this equation $3x+3^{\ln{x}}-4=0$

Please help me to solve this equation $$3x+3^{\ln{x}}-4=0$$ I found $x = 1$ in a plot W.A, but can I find x by Algebraic solution? Thanks
1
vote
1answer
45 views

Calculating percentage basics

I am new to this website, so please forgive my mistakes if there is any. I'm not quite good standing with mathematics and I have trouble finding the solution for this problem: 60% of the number 50 ...
3
votes
8answers
318 views

Real numbers in math

What are real numbers for a person who doesnt know ANYTHING about math, and had to explain them what real numbers are. Are real numbers only rational and irrational? if so then do we have to say what ...
3
votes
1answer
125 views

Show that $a^x+b^x+c^x>(a+b+c)^x$ for all $a,b,c>0$ and $0<x<1$

Hy all! I'm having trouble finding a proof for the following problem: Show that $a^x+b^x+c^x>(a+b+c)^x$, if $a,b,c>0$ and $0<x<1$ (over the real numbers). This inequality has been ...
1
vote
0answers
56 views

Arithmetic series question? [duplicate]

Hello can anyone help me with question. In an arithmetic series the terms of the series are equally spread out. For example in $1+5+9+13+17$ consecutive terms are $4$ apart. If the first term in an ...
5
votes
7answers
494 views

What is the value of $i+i^2+i^3+\cdots+i^{23}$? [duplicate]

Can anyone help me with this question and show me a step by step solution please? The imaginary number is $i$ is defined such that $i^2=-1$. What is $i+i^2+i^3+\cdots+i^{23}$?
1
vote
2answers
1k views

Formula for Snake Draft pick numbers

Hello I am trying to come up with a formula to calculate the overall pick number in a snake style draft. For example in a snake draft every other round the pick order reverses. So in a 10 team league ...
0
votes
2answers
310 views

Sketching the graph of $\ln(4-x)$ by transforming $\ln(x)$

Using the base graph of $\ln(x)$: I "reflect" the graph of $\ln(x)$ in the $y$-axis, giving me $\ln(-x)$, the graph now cuts the $x$-axis at $(-1,0)$. I then translate the graph by negative $4$ ...
4
votes
1answer
63 views

How can the formula be found for this problem?

We have a truck that we need to completely fill up with merchandise. We have an infinite supply of merchandise of dimension $1\times1\times1, 2\times2\times2, 4\times4\times4, 8\times8\times8, ...
3
votes
1answer
77 views

How to find more numbers like this?

We have the number 153, which has the following special property: $$153 = 1^3 + 5^3 + 3^3$$ How can we find more numbers like this mathematically (so without making guesses (or even educated ...
3
votes
1answer
73 views

Inequality problem $(a^2-b^2)(a^4-b^4)\le (a^3-b^3)^2$

This problem that I have been unable to solve is from the book "Introduction to Inequalities" by Beckenbach and Bellman, chapter 2, page 22, problem 4. Problem 4. Show that $$(a^2-b^2)(a^4-b^4)\le ...