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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2answers
27 views

How to Simplify Sin/tan problem.

I am trying to simplify $\displaystyle\frac{\sin^2}{\tan^2}$ but I don't know how to go about it. Any help is appreciated.
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1answer
10 views

Interesting association between tangent lines of slope one and ellipses

Why is it that a tangent line with slope $1$ to an ellipse centered at the origin will have a transformation of $\pm \sqrt{a^2 +b^2}$ where $a$ and $b$ are the major and minor axis of the ellipse? ...
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2answers
23 views

Proving that function with domain (-1,1) is injective.

Function $g\colon (-1,1) \rightarrow \mathbb R$ is defined by $g(x)=\dfrac{x}{1-x^2}$. Prove that $g(x)$ is injective. Work: I shifted the equation so that it ends up like ...
3
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1answer
32 views

Calculate depth using triginometry

I was asked a question like this on an exam today and I'm wondering if I got it right or not. ...
2
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0answers
32 views

Understanding underlying algebra for calculus convergence problem

I'm working on series convergence/divergence problems in my Calc 2 class, and (as has happened often), I'm hung up on some underlying algebra. The first step in the solution manual for a problem I'm ...
2
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4answers
32 views

Calculate the height of a building

This question I really need help with, I simply do not know where to start! Anyone can help, all I can offer is supreme thanks. Please include method. I don't want simple answers which don't help me ...
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0answers
12 views

Substitution in multiple integrals, rewriting variables

I have this problem that seems almost laughably simple, but has stumped me for some time. For the variables $ u = x^2 - y^2 $ $ v = x$ $y$ Under the condition $ u > 0 $ We're simply asked to ...
3
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5answers
82 views

How do I show that $\sqrt{5+\sqrt{24}} = \sqrt{3}+\sqrt{2}$

According to wolfram alpha this is true: $\sqrt{5+\sqrt{24}} = \sqrt{3}+\sqrt{2}$ But how do you show this? I know of no rules that works with addition inside square roots. I noticed I could do ...
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2answers
25 views

The zoo car park

In the zoo car park there were 780 vehicles. Most of the vehicles were cars,but one section contained coaches only.the ratio of cars to coaches was 11:2. How many vehilces were cars and how many were ...
2
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2answers
33 views

Question about $I(n)=x^n \cos x$

If $I(n)=x^n \cos x$, find $I(n)$ in terms of $I(n-2)$. This is a reduction formula question.
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2answers
54 views

Logarithm of a negative number

We know this identity: $\ln(\frac{a}{b}) = \ln(a) - \ln(b)$ Suppose both $a$ and $b$ are negative. Then the left-hand size evaluates to something, it is a logarithm of a positive number (minus signs ...
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3answers
34 views

inverse of a function f(x)..change x and y

Find the inverse of the function $f(x)= (2x-1)/(x^2-1).$ we switch the x and y letters and then solve the the equation...but it became kind of complicated while solving
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0answers
42 views

Don't know when to add negative numbers

I'm definitely not a math person and only did general mathematics in high school, and unfortunately, not paying as much attention to that as I should have. Well, I'm doing Discrete Mathematics in my ...
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11answers
1k views

How to solve $4\sin \theta +3\cos \theta = 5$

Another problem that I already wasted hours on. Given $$4\sinθ +3\cosθ = 5$$ Find $$4\cosθ -3\sinθ$$ Help me guys (PS:I'm not that good in maths)
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1answer
16 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 ...
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1answer
37 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
8 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
21 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 ...
0
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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
20 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) ...
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1answer
19 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
52 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
49 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
17 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 ...
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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 ...
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7answers
111 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
41 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 ...
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1answer
39 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
31 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 ...
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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.
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2answers
59 views

Can you only take positive numbers into a square root?

I am thinking about when you are simplifying something like this $x(1/x)^{0.5}=x^{0.5}$ but if $x$ was negative then you have lost the sign why is this still valid or is it not?? Another example is ...
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1answer
38 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 ...
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3answers
69 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.
0
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2answers
35 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
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3answers
38 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 ) ...