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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6
votes
5answers
264 views

Solving an equation with fractional powers

I was trying to find the maximum value for a function. I took the first derivative and arrived at this horrible expression: $$ (x^2 + y^2)^\frac{3}{2} - y {\frac{3}{2}}(x^2 + y^2)^{\frac{1}{2}}2y = ...
0
votes
3answers
90 views

Arithmetic and Geometric Sequences (with already-known ratio/difference)

Arithmetic sequence: common difference = $10$ Geometric sequence: common ratio = $2$ Arithmetic: $f(1) = 50$ $f(2) = 60$ $f(3) = 70$ $f(4) = 80$ ...
7
votes
1answer
187 views

Closed-form expression for $\sum_{n=1}^{k} (-1)^{n+1}n^2(n^2-1)\binom{2k}{k-n}$?

Wolframalpha tells me that $$\sum_{n=1}^{k} (-1)^{n+1}n^2(n^2-1)\binom{2k}{k-n}=0$$ for $k>2$ However I have not been able to come up with a proof and I simply don't see how to do it. Does anyone ...
1
vote
1answer
381 views

Algebra 2 Completing the square: Find the dimensions

A rectangular swimming pool is $6$ ft. deep. One side of the pool is $2.5$ times longer than the other side. The amount of water needed to fill the pool is 2160 cubic feet. Find the dimensions. Do ...
2
votes
4answers
433 views

Filling a swimming pool with two hoses

Hose A can fill a pool in 4 days. You add a second hose B, now both A and B can fill it in 3 days. How many days will it take to fill the pool with just hose B? I would like to see a formula.
0
votes
3answers
113 views

A fund of $30,000 is used to award scholarships…If i=0.09, find the number of scholarships which can be awarded

A fund of $30,000 is used to award scholarships of amount 3000, one per year, at the end of each year for as long as possible. If i=0.09, find the number of scholarships which can be awarded and the ...
1
vote
2answers
175 views

Problem with graphing linear equations

Well, I can understand how to graph basic liner equations, for example: $$y=2x-4$$ The y-intercept would be -4 and the slope would be 2. The coordinates could then be (0,-4)(1, -2) However, how ...
0
votes
0answers
57 views

Convert function $g(x) = \frac{\sqrt{2+x}}{x-3} $ to piecewise sub functions

I know that the domain is $[-2,3)\cup (3, \infty)$, but I don't know how to describe the functions in each section of the domain. Thanks, Steve
2
votes
6answers
995 views

Polynomials with integer coefficients

Through definitions, theorems and my professor the following is true: The product of any two odd integers is odd. The sum and difference of any two odd integers are even. The sum, product and ...
1
vote
1answer
57 views

$|px^2+qx+r|\le1$ for all $x$ in $[-1; 1]$, show that $|rx^2+qx+p|\le2$ for all $x$ in $[-1; 1]$

I have an interesting problem, in which i would be utmost happy to receive some help: $p$, $q$, $r$ are real numbers with the following property: $|px^2+qx+r|\le 1$ for all $x$ in $[-1; 1]$. Prove ...
-4
votes
2answers
773 views

What happens as you take repeated square roots, starting with 8? What does the answer approach as you take more and more square…

a) What happens as you take repeated square roots, starting with 8? b) What does the answer approach as you take more and more square roots? c) Would the answer be the same if you started with any ...
2
votes
4answers
285 views

The number of real roots of $(x+3)^4 + (x+5)^4 = 16$

I was solving some problems and I came across this question: Q: The number of real roots of $(x+3)^4 + (x+5)^4 = 16$ is ...
3
votes
1answer
92 views

How can I simplify $\frac{\sqrt{x} + 1}{x\sqrt{x} + x + \sqrt{x}} : \frac{1}{x^2-\sqrt{x}}$?

$$\frac{\sqrt{x} + 1}{x\sqrt{x} + x + \sqrt{x}} : \frac{1}{x^2-\sqrt{x}}$$ As I'm trying to study calculus so I will be thankfull to just a hint, not full solution. Thanks.
0
votes
1answer
55 views

I can solve this but how can you write an expression for it that will work with any number? [duplicate]

A company charges 8.00 to dig a hole one foot deep. The second foot is 5.40, the third $4.32, etc. (Note that the cost of each subsequent foot is 20% less than the previous). The question asks what ...
0
votes
1answer
244 views

adding two quotients where the single-variable is in the denominator of both, yet with diferent coefficients

Here is the equation I'm trying to solve for Physics: $f(x) = \frac {K}{x^2} - \frac {K}{2(x-4)^2} = 0 $ Where $K$ is just some constant and $x$ is the single variable. I've got to find the x value ...
1
vote
0answers
31 views

Sum of spatio-temporal window

I would like to get confirmation about a formula. It looks quite simple but I can't make it work, so I start to have some doubt. Let's say I have 2 images $I_t$ and $I_{t-1}$ which are 2 consecutives ...
3
votes
2answers
308 views

How to factor $a^n - b^n$?

Wikipedia provides a proof, but I don't understand how: $$a^n - b^n = (a-b)(a^{n-1} + ba^{n-2} +\cdots + b^{n-1})$$ follows from $$x^{n-1} + x^{n-2} +\cdots + x + 1 = \frac{x^n - 1}{x-1}$$ Could ...
1
vote
1answer
33 views

Calculate how long it take to reach a goal

Given a growth rate for a period and a goal. How can I calculate how many periods that it will take to reach that goal. Example: An investment currently valued at $400 grows at 30% per week. ...
2
votes
1answer
3k views

What are the most famous (common used) precalculus books and its differences?

I'm trying to decide which one to pick up to begin a self study of mathematics. One of the factors is how much content is covered and the amount of associated solved problems the book has. EDIT: ...
5
votes
6answers
386 views

High School Mathematics

Could you recommend any high school mathematics books that are rigorous and present high school mathematics in a higher level.I just realized that I only memorized a lot of things that I was taught in ...
0
votes
2answers
1k views

Root of a polynomial with rational coefficients

I am currently learning about Direct Proofs. I am struggling trying to find a starting point to prove the Statement: For all real numbers $c$, if $c$ is a root of a polynomial with rational ...
0
votes
2answers
1k views

Help with finding all the roots to $z^6 - 2z^3 + 2 = 0$.

I need help with finding the roots to the equation $z^6 - 2z^3 + 2 = 0$ I start with assigning $x$ as $z^3$. This gives me the equation: $x^2 - 2x + 2 = (x-1)^2 + 1 = 0$. Further developments: ...
1
vote
4answers
565 views

Find the value of the following expression

I came across the following problem in an Exam that says: Find the value of the expression ...
0
votes
3answers
553 views

How to solve $z^2 - (2 + 2i)z - 5 -10i = 0$?

I am trying to solve the following equation: $z^2 - (2 + 2i)z - 5 -10i = 0$? My attempts so far has been trying to complete the square of $z^2 - (2 + 2i)z - 5 -10i$ but I have had no real progress ...
0
votes
1answer
48 views

Playing scales on the piano with both hands at different rates

Say I am playing both bass and treble on the piano. With my right hand, every time I reach the next C up, I walk my fingers back down to middle C: |Cm|D|E|F|G|A|B|C1|B|A|G|F|E|D|Cm| or ...
2
votes
1answer
160 views

Property true for some integers and false for others: $-a^n$ = $(-a)^n$

I am currently working in my Discrete math class with elementary number theory and methods of proof. I have been given the problem $-a^n = (-a)^n$. According to the professor and the book this ...
0
votes
2answers
49 views

Show that $7(3(2)^k + 2(5)^k) - 10(3(2)^{k-1} + 2(5)^{k-1}) = 3(2)^{k+1} + 2(5)^{k+1}$

$7(3(2)^k + 2(5)^k) - 10(3(2)^{k-1} + 2(5)^{k-1}) = 3(2)^{k+1} + 2(5)^{k+1}$ The problem is part of a proof. If you could also talk me through your thought process for solving this problem, I would ...
1
vote
2answers
4k views

Finding the coordinates of the point where each line crosses the y-axis

I have a problem like this: Give the coordinates of the point where each line crosses the y-axis. Then it gives me an equation in slope-intercept form, here is ...
1
vote
0answers
113 views

A construction of the trig functions on the unit circle

Can anyone shed some light on this picture? http://upload.wikimedia.org/wikipedia/commons/9/9d/Circle-trig6.svg I am not interested in "$\sin$", "$\cos$", or the outdated trig functions. How do we ...
0
votes
1answer
72 views

Show that $6(5^k) - 6 - 5^k + 5 = (6-1)5^k -1$. Explain it to me like I'm 5 please…

I'm studying for a discrete math course, and I'm finding out that I'm really weak in algebra. I don't see how this step, $6(5^k) - 6 - 5^k + 5 = (6-1)5^k -1$ happened in a proof I'm looking at. ...
3
votes
4answers
1k views

How to rewrite $\sin^4 \theta$ in terms of $\cos \theta, \cos 2\theta,\cos3\theta,\cos4\theta$?

I need help with writing $\sin^4 \theta$ in terms of $\cos \theta, \cos 2\theta,\cos3\theta, \cos4\theta$. My attempts so far has been unsuccessful and I constantly get developments that are way to ...
0
votes
1answer
246 views

Indices Again - Expressing Negative Fraction Powers as powers of given number

If someone could provide me with a website or link with information about this I'd be grateful but otherwise: I have the number two (with the ability to add a power) and the number ...
3
votes
4answers
398 views

Help with showing how $\sin\alpha\cos\beta$ $=$ $\frac{1}{2}(\sin (\alpha + \beta) + \sin(\alpha-\beta))$ using Eulers formula.

I need help with understanding how one can rewrite: $\sin\alpha\cos\beta$ to be equal to: $\frac{1}{2}(\sin (\alpha + \beta) + \sin(\alpha-\beta))$ using Eulers formula. I know that it probably ...
1
vote
2answers
1k views

Need help using De Moivre's theorem to write $\cos 4\theta$ & $\sin 4\theta$ as terms of $\sin\theta$ and $\cos\theta$

I need help with the following question: "Use De Moivre's theorem to write $\cos 4\theta$ & $\sin 4\theta$ as terms of $\sin\theta$ and $\cos\theta$" You could write the problem as: ...
1
vote
2answers
55 views

Absolute value inequality - Please guide further

Prove that if the numbers $x$, $y$ are of one sign, then $|\frac{x+y}{2}-\sqrt{xy}|+|\frac{x+y}{2}+\sqrt{xy}|=|x|+|y|$. Expanding the LHS, $$\left|\frac{x+y}{2}-\sqrt{xy}\right|= \left|\frac{x+y ...
3
votes
3answers
95 views

Calculation of polynomial $g(x)$ satisfies $x\cdot g(x+1)=(x-3)\cdot g(x)$

If a polynomial $g(x)$ satisfies $x\cdot g(x+1)=(x-3)\cdot g(x)$ for all $x$, and $g(3)=6$, then $g(25)=$? My try: $x\cdot g(x+1)=(x-3)\cdot g(x)$, Put $x=3$, we get $g(4)=0$, means $(x-4)$ is a ...
2
votes
1answer
66 views

How to prove this inequality : $(x^a+y^a)^{\frac1{a}}>(x^b+y^b)^{\frac1{b}} \, ; x>0,\ y>0;\ 0<a<b$

Prove that when $\displaystyle x>0,\ y>0;\ 0<a<b$ $$\displaystyle(x^a+y^a)^{\frac1{a}}>(x^b+y^b)^{\frac1{b}}$$
1
vote
2answers
42 views

Creating an application with a level system and I would like to convert the values into an equation

Here is what I have so far: you start at level $0$ with $0$ XP. The objective is to gain XP to level up. Once you reach $100$ XP you get to level $1$, $300$ XP = Level $2$, $600$ XP = Level $3$, ...
2
votes
4answers
456 views

Common multiple question?

How would I solve the following question find the least common multiple of these two expression. $14w^7y^2$ and $6w^4y^5x^8$ would I just have to multiply them.
2
votes
1answer
35 views

Determining constant values from 3 equations

I have the following three equations: $$\begin{align*} k_1 + k_3 &= 0\\ k_1e^{k_2(0.1)} + k_3 &= 1\\ k_1e^{k_2(1)} + k_3 &= 100 \end{align*}$$ How do you go about solving for values ...
1
vote
3answers
82 views

basic rules logarithm of exponential

I am looking for proof of the basic rules of logarithm. I can prove all basic rules except this $$\log_ab^y=y\log_ab$$ how to get this rule using definition of logarithm.
2
votes
4answers
144 views

Prove without induction : $\sum_{k=1}^{2n} \frac{(-1)^{k+1}}{k} = \sum_{k=1}^n \frac{1}{k+n}$

Prove without induction that : $$ \sum_{k=1}^{2n} \frac{(-1)^{k+1}}{k} = \sum_{k=1}^n \frac{1}{k+n} $$ Please if you have any elementary tricks just post hints.
2
votes
2answers
160 views

Non-induction proof of $2\sqrt{n+1}-2<\sum_{k=1}^{n}{\frac{1}{\sqrt{k}}}<2\sqrt{n}-1$

Prove that $$2\sqrt{n+1}-2<\sum_{k=1}^{n}{\frac{1}{\sqrt{k}}}<2\sqrt{n}-1.$$ After playing around with the sum, I couldn't get anywhere so I proved inequalities by induction. I'm however ...
2
votes
1answer
2k views

Perpendicular line passing through the midpoint of another line

I have several $2d$ line segments. for example, if I take a one line segment having end points $(x_1, y_1)$ and $(x_2, y_2)$. Then, I want to make a perpendicular line which passes through the ...
1
vote
2answers
139 views

Integer solution of $x^8-24x^7-18x^5+39x^2+1155=0$

The number of Integral Roots of the equation $x^8-24x^7-18x^5+39x^2+1155=0$ My Try: Using integral roots Theorem, integer solution of this equation is all possible factor of $1155 = \pm 3 ...
-1
votes
1answer
63 views

Why $\lim_{x\to a}\frac{x-a}{x^2-a^2}=\frac{1}{2a}$?

Given that $a\neq0$, what is the value of the following limit: $$\lim_{x\to a}\frac{x-a}{x^2-a^2}$$ I know the answer is $\frac{1}{2a}$ but why? If I substitute $a$ into the equation I get an ind of ...
4
votes
4answers
315 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
51 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
152 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 ...