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
2answers
432 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
34 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
4k 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
437 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
755 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
708 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
51 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
162 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
5k 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 an example: $y=3x+4$ Would I ...
1
vote
0answers
117 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
75 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
311 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
427 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
2k 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
64 views

Absolute value inequality - Please guide further

Prove that if the numbers $x$, $y$ are of one sign, then $\left|\frac{x+y}{2}-\sqrt{xy}\right|+\left|\frac{x+y}{2}+\sqrt{xy}\right|=|x|+|y|$. Expanding the LHS, ...
3
votes
2answers
99 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
69 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
47 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
1k 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
91 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
147 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
179 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
3k 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
144 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
488 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
55 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
163 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
363 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
191 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
92 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
179 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
105 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
223 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
344 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
227 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
134 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
31 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
218 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 ...