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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1answer
13 views

How to handle a double inequality where all 3 spots have unknowns

Given the problem $4x \lt 2x + 1 \le 3x + 2$ solve for x. I'm not sure how to go about solving this problem. No matter how I subtract or add the x's or multiply/divide the coefficients I cannot ...
4
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1answer
35 views

Need help solving exponential equation $2\mathrm{e}^x=5-\mathrm{e}^{-x}$

I need help solving $2\mathrm{e}^x=5-\mathrm{e}^{-x}$. I've tried many ways of solving it but I keep getting the wrong answer. By the way, my book says the solutions are $x=-1.518$ and $x=0.825$ ...
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1answer
7 views

Insert Means in an Arithmetic Sequence (that contains logarithms)

So the question is: You have an Arithmetic Sequence. Log 2 and Log 1024 are two terms in the sequence Find 8 arithmetic means between them.
2
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1answer
31 views

Correct equation for this question

The question is, A small hydroelectric generating station can produce 17 MWh of energy in 12 months. AFter 4 months of operation, another generator is added. This additional generator can produce 11 ...
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0answers
16 views

the two travelers

Two travelers, Manuel and Brooke, set out from two places, Manhattan and Brooklyn, and at the same time. Manuel from Manhattan with a design to pass through Brooklyn, and Brooke from Brooklyn to ...
1
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1answer
27 views

What are $a$ and $b$ when the zeropoints of $f(z)=(a+bi)z+2-i=0$ is at $1-i$?

$f(z)=(a+bi)z+2-i$. What are the values of a and b when $1-i$ is the zeropoint of f? $f(z)=(a+bi)z+2-i=0$ $(a+bi)(1-i)+2-i=0$ $a+bi-ai-bi^2+2-i = 0$ $(a+b+2)+(-a+b-1)i=0$ I don't know what the ...
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1answer
17 views

Velocity problem

Beni and Gal compete in running on straight track AB, that distance is 50m. They both started from point A on that track and run to point B. Beni leaped first. Gal leaped one second after Beni and ...
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1answer
13 views

System of equations - lagrange

Solve this system of eq: $$\begin{cases} yz &= 2 \lambda x &\,\,\,(a)\\ xz &= 2 \lambda y &\,\,\,(b)\\ xy &= 2 \lambda z &\,\,\,(c) \\ x^2+y^2+z^2−3&=0 &\,\,\,(d) ...
0
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1answer
16 views

How to calculate Polar coordinates for Complex Polynomials of Higher Degree?

When such I have a complex number such as $3 - 4i$, I can calculate the $r$ with $r=\sqrt{X^2+Y^2} = \sqrt{3^2+4^2}$. But how do I solve this when I have a complex number such as $(2+6i)^6$
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1answer
51 views

How to quickly determine which number is bigger than the other?

$$\{1,2,66,99\}\cup\{5,7,9\}=\{1,2,5,7,9,66,99\}$$ But if I have for example $$\{0,2,2\sqrt{2},2\sqrt{3}\}\cup\{\sqrt{5},\sqrt{7},2\sqrt{9}\}$$ Should I first find whose element is bigger than the ...
2
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1answer
47 views

How can I evaluate this sum of product?

$$\sum_{n=1}^{\infty}\prod_{k=1}^n\left(\frac{\sqrt{k+1}-1}{\sqrt{k}}\right)$$ I have no idea . Thank you.
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0answers
16 views

Word velocety problem

There is a track AB with length of 50 meters. The first runner, Sam starts from A to B, after one second the second runner, Jim, starts running, and after 10 meters they meet. When Jim arrives to ...
0
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1answer
34 views

Solve a system of inequalities

$$\begin{cases} \log_{2}^{2}(-\log_{2}x) + \log_{2}\log_{2}^{2}x \leq 3 & \\-4 |x^2-1|-3\geq \frac{1}{x^2-1}& \end{cases}$$ What I've tried: Make substitution $t=x^2-1$ and solve second ...
5
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7answers
105 views

Find value range of $2^x+2^y$

Assume $x,y \in \Bbb{R}$ satisfy $$4^x+4^y = 2^{x+1} + 2^{y+1}$$, Find the value range of $$2^x+2^y$$ I know $x=y=1$ is a solution of $4^x+4^y = 2^{x+1} + 2^{y+1}$ , but I can't go further more. I ...
0
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2answers
39 views

Simple question on exponentiation

I know this one is trivial, but I was wondering: if I have something like $$a^{b^c}$$ then i know that it should be read as $$a^{\left(b^c\right)}$$ if no other parenthesis is present. Question: if ...
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4answers
77 views

Finding $a$ and $b$ from $a^3+b^3$ and $a^2+b^2$

Question 1 Two numbers are such that the sum of their cubes is 14 and the sum of their squares is 6. Find the sum of the two numbers. I did $a^2+b^2=6$ and $a^3+b^3=14$ Find $a$ and $b$, two ...
0
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1answer
38 views

Proving that $\frac 1 2 t^3+\frac 1 2 t, (t\in \mathbb N)$ will always be a whole number

Is there any way to show that $\dfrac 1 2 t^3+\dfrac 1 2 t$ will always be a whole number? Assuming that t is always a whole number greater than or equal to one? If i think about it logically it seems ...
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2answers
36 views

the geometric explain of $t = x-\frac{a}{3}$ in the simplify of cubic equation $x^3+ax^2+bx+c=0$

Assume $$f(x) = x^3+ax^2+bx+c$$ we have $$f''(x)=2a+6x$$. we get $x = -\frac{a}{3}$ Magically, If we take the transformation: $$t = x -\left(-\frac{a}{3}\right)$$. we can transform the above ...
3
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2answers
52 views

Solutions for quartic

Suppose I have an equation in the form $(x-a)^4 + (x-b)^4 = c$. What is a clever way to find all four solutions? I have tried expanding and then used long division. However, I believe a better way is ...
0
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4answers
33 views

Equation of a line

I've been practising some past previous exam question and came across this: How would you find the equation of the line with y-intercept? -$\frac{3}{4}$ and slope -$\frac{1}{2}$?
0
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4answers
58 views

Prove that (integer + non-integer) never equals an integer.

My question is how do you prove that given an integer $x$ and a number $y$, the only way for $x + y$ to be an integer is if $y$ is also an integer. I can see how to prove by induction that an integer ...
0
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1answer
19 views

Finding the value of trigonometric functions

This is probably one of the easiest concepts but I do not get it, so I am going to give the two problems that are giving me the most trouble on my very long worksheet I have to do, maybe you guys can ...
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1answer
27 views

a question on time, distance and speed

Ram and Gopi run towards each other starting from K and L respectively with respective speeds of 2 kmph and 3 kmph. After meeting each other, to reach L, if Ram takes 7 hours less than the square of ...
1
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1answer
27 views

Routes to a house

In this city, all the streets that run North and South have lettered names (A,B,C, etc.) and all the streets that run East-West have numbered names (1st, 2nd, 3rd, etc.). As you drive East, the ...
1
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3answers
48 views

Solve the System of Equations in $x$ and $y$

\begin{equation} x+\frac{3\,x-y}{x^2+y^2}=3 \tag{1} \end{equation} \begin{equation} y=\frac{x+3\,y}{x^2+y^2} \tag{2} \end{equation}
2
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3answers
53 views

How do I solve $(x^3-4x^2+5x-6)/(x^2-x-6)=4$ algebraically?

How do I solve $(x^3-4x^2+5x-6)/(x^2-x-6)=4$ algebraically? I tried: $4(x^2-x-6)=x^3-4x^2+5x-6$ $4x^2-4x-24=x^3-4x^2+5x-6$ $x^3-8x^2+9x+18=0$ I don't know how to solve this algebraically.
1
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1answer
36 views

How do I solve $(x^3-4x^2+5x-6)/(x^2-x-6)$ using long division?

I tried to solve $(x^3-4x^2+5x-6)/(x^2-x-6)$ using long division... $x^2-x-6$ / $x^3 -4x^2+5x-6$ \ $x-3 $ $\underline{x^3 - x^2 - 6x}$ - $-3x^2$ $\underline{-3x^2-3x-18}$ - $4x+12$ I made ...
1
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3answers
68 views

Solution for $x$ with exponents?

I am trying to solve the following, $$7^{(2x+1)} + (2(3)^x) - 56 = 0$$ Should I put the 56 on the other side and get the log of both sides and is there a better way to solve this.
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2answers
34 views

How to work out which of these two equations is larger?

I have two equations: $x = 2^n (p+i)^{3n}$ $x = 14^n p^{3n} $ Here $n$, $p$, and $i$ are all integers $\geq0$. I worked out (using a spreadsheet) that if $i > 1$ then the value of x in ...
3
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2answers
70 views

If $(b-a) = (ba^2 - ab^2)$, show that $a=b$ where $a,b \in \mathbb R$

I am slightly stuck on this seemlingly simple problem that I encoutered as part of a problem to show that the orthogonality condition of $M_{2\times2}$ matrices given by $\sum_i a_{ij}a_{jk} = ...
2
votes
3answers
37 views

Simplifying Simultaneous equations

I'm doing an 8th grade math text book, and came across this problem: $4a + $ $ 5b + $ $9c$ = 36; and 7a + 9b + 17c = 66, what is a + b + c = ? The first thing I could notice was that, " How could ...
1
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1answer
15 views

population of a city based on another [hw]

I've been struggling with understanding how to solve a particular branch of population problems. The question reads as follows: City P had a population of 20,000 in the year 2005. The population ...
1
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3answers
28 views

Solving for the equation, giving exact solutions between (0 and $2\pi$)

I have the problem: $\sin(2x)=\tan(x)$ I used the double angle formula to get $2\sin(x)\cos(x)=\tan(x)$. But after that step, I do not know whether or not to subtract $\tan(x)$ or to set ...
0
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4answers
33 views

Solving a logarithmic system of equations

I am working on a test study guide and I can't seem to get the correct answer for this system of equations: \begin{align*} \ln(x) &= 3\ln(y) \\ \ 3^x &= 27^y \end{align*} I'm not ...
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1answer
33 views

Solving for a Sine equation

I have the equation $ \ \sin(2x-1) \ = \ 0 $ . I know the answer is $ \ .5 + \pi k \ $ . but I don't understand how the answer was derived if the domain for sine must be between ($ -\pi/2 , \pi/2 ) ...
0
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1answer
18 views

Inequality involves complex numbers

Show that if $|arg(z)| \leq \frac{\pi}{4}$ then $ x \geq 0 $ and $|z| \leq \sqrt{2}x$, where $z=x+iy$. My question is : How can I prove that x is greater than 0 and the other is, can I use the fact ...
0
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2answers
35 views

Solving the trigonometric equation $\cos(x)-.4=0$.

I need to find the exact equations (if possible) for $\cos(x)-.4=0$. I have found the first answer to be $1.159+2\pi k$. For the second answer, do I have to subtract $2\pi$ and $1.159$ to get the ...
0
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2answers
17 views

Perimeter algebraic question

I am so out of touch with my Maths! I am trying to help my daughter with her Maths, so any help would be truly appreciated. Thank you very much in advance. Question: The perimeter of rectangle A ...
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4answers
78 views

Why all such polynomials have $-1$ as a root?

Why all polynomials of this form have $-1$ as a root? $ x^5+x^4+x^3+x^2+x+1 $ and similar polynomials like $ x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1$
0
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0answers
7 views

Re-arrange formula to solve for variable [on hold]

Create formula to solve for b. (I^(b)(I^(1-b)-J^(1-b)))/((1-b)(1/b)((1-K)^(-b)-1)(1/12)) I'm so lost haha any help would be greatly appreciated. Best, G
0
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1answer
21 views

What are a,b, and c in $(3)/(x^3+ax^2+bx+c)$?

What are a,b, and c for the function $(3)/(x^3+ax^2+bx+c)$ with asymptotes at x=0, y=0, x=2 and x=4? I got stuck when I tried to calculate $2^3+2^2a+2b+c=4^3+4^2a+4b+c$. $8+4a+2b+c=64+16a+4b+c$. ...
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0answers
6 views

How to normalize one number to another

Lets say I have some data like Number 1 Number 2 CombinedResult 2.5 512 1280 2.1 60 126 1.5 3000 4500 I ...
3
votes
3answers
170 views

Rationalization of denominator with nth roots

I would need your help to solve a problem with the rationalization: \begin{equation} \sqrt{a}-\sqrt{b}\over \sqrt[4]a+\sqrt[4]b \end{equation} I think I'm doing something wrong with the products and ...
1
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2answers
32 views

Given a formula with variables, how can I turn it around to find another solution

First of all, sorry for the crappy title, I have no idea how to ask this question. I have a formula: $$S = \left(\frac{T\cdot D\cdot C}{1000}+V^{\frac13}\right)^3$$ I now need to turn it around to ...
3
votes
1answer
51 views

How to evaluate this sum 2?

$\displaystyle\sum_{x+y+z=2014}xy^2z^3$ $\quad , x,y,z\in\mathbb{N}$ I think it maybe use combinatorial method.
0
votes
1answer
79 views

If $x + \frac{1}{x} = k$, what's the value of this sum?

Friends, if $x + \frac{1}{x} = k$, with $k$ positive real number, what's the value of $$1+ x^6 + x^{12} + x^{18} + x^{24}+x^{30}$$ I tried with the substitution $u= x^6$: $$a= 1+ u + u^2 + u^3 + u^4 ...
0
votes
1answer
23 views

Absolute value question when solving with square root function

Given the equation $x + (y-1)^2=0$ and solving for $y$ I obtain $\vert y-1\vert = \pm\sqrt{-x}$. I know that the final answer is $y = 1 \pm \sqrt{-x}$, but what I don't understand is why taking off ...
0
votes
3answers
31 views

Question about variable and constant notation in some properties

I am just starting to think about mathematical notation a lot more and some parts of it do not make as much sense to me as I would like. I am operating under the convention that beginning alphabet ...
0
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0answers
39 views
0
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0answers
32 views

Finding the inverse of trig functions

I'm supposed to find the inverse of $$f(x) = \cos(x)+x$$ I usually just substitute $x$ for $y$ and then re-arrange. What do I do in this scenario?