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
98 views

Upper approximation of $\mathrm{atanh}(x)$?

Is there are nice upper approximation of $\mathrm{atanh(x)}$? For example, $\ln(x)$ is nicely approximated by $x-1$ for $x$ around $1$.
1
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1answer
83 views

Technique for solving $ x^4 - x^3 + x - 1 = 0 $

Here's another idiotic algebra question that I can't seem to make any progress on. $$ x^4 - x^3 + x - 1 = 0 $$ I tried to make it into a quadratic: Let $ u = x^2 $, then $$ u^2 - xu + x - 1 = 0 $$ ...
0
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2answers
50 views

Is it possible to do this? Write a fraction as a product

I have two quantities $A$ and $B$ and I consider the fraction $$\frac{1}{A+B}$$ I would like to write the above expression as a Product, i.e. find functions $F$ and $G$ such that $$\frac{1}{A+B} = ...
1
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2answers
184 views

Arithmetic progression on finding first term and common difference

How to do with the question ask to find out the values of a= first term and d=common difference if the sum of the first four term is equal to three times the fourth term and the eighteenth term is 3? ...
3
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4answers
129 views

Why is it impossible to find natural numbers $a$ and $b$ such that $12b^2=a^2$?

This was a question in the exercises for an EdX course by Prof Starbird on Effective Thinking through Mathematics which was long over, but I am working through the course at my own pace. I feel that ...
2
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2answers
47 views

Finding a specific term in an expansion $(a+b)^n$ without expanding

How can I find a term within an expansion without actually expanding or using Pascal's Triangle? For example: 5th term of $$ \left(\dfrac{x}{y}-\dfrac{y}{x}\right)^8 $$
1
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2answers
33 views

Help with significant figures unit conversion…

So I am having trouble remembering the trick to convert from square metres etc. to square millimetres. Say I have $2\cdot{10}^{-3}m^2$ and I want to get it into millimetres. I vaguely remember ...
13
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10answers
4k views

Taking Calculus in a few days and I still don't know how to factorize quadratics

Taking Calculus in a few days and I still don't know how to factorize quadratics with a coefficient in front of the 'x' term. I just don't understand any explanation. My teacher gave up and said just ...
1
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1answer
61 views

Solving simultaneous equations with `min{}` function

I have following system of m number of simultaneous equations with min{} function. These equations are symmetric as well. ...
1
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3answers
88 views

What are the steps to solving |3x + 1| > |2x - 7| with the given answer as $(-∞,-8)\cup(6/5,∞)$?

What are the steps to solving $|3x + 1| > |2x - 7|$ with the given answer as $(-∞,-8)\cup(6/5,∞)$? I am having difficulty with understanding inequalities with absolute value functions on both ...
2
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1answer
162 views

Simplify the radical

I need to simplify this radical, $\sqrt{2+e^{8t}+e^{-8t}}$ How is this done? I do not know where to go from here to simplify this further.
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1answer
228 views

To Find the height of the building

A building casts a shadow 50 feet long. A rod 4 feet tall placed near the building casts a shadow 3 inches long. Find the height of the building.
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5answers
147 views

Complex solutions to $ x^3 + 512 = 0 $

An algebra book has the exercise $$ x^3 + 512 = 0 $$ I can find the real solution easily enough with $$ x^3 = -512 $$ $$ \sqrt[3]{x^3} = \sqrt[3]{-512} $$ $$ x = -8 $$ The book also gives the ...
1
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0answers
77 views

Proving uniqueness

Sorry this is such a trivial question, but I'd like to check my understanding on this subject before proceeding. Suppose that we have the quantity $x+a=b$ where $x$ is a variable, and we wish to show ...
1
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2answers
103 views

How to simplify $\sqrt{\sqrt{5}+1} \cdot \sqrt{\sqrt{5}-1}$?

This is the original problem: $\sqrt{\sqrt{5}+1} \cdot \sqrt{\sqrt{5}-1} = x$. I'm really confused about how to solve this problem, I come as far as saying this: $\sqrt[4]{5} + \sqrt{1}\cdot ...
0
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1answer
29 views

Doing wrong in this fraction simplification?

$$ \frac{5}{2x-3} - \frac{3}{(2x-3)^2} $$ I have to simplify So I had the minimun common multiple in $$(2x-3)^2$$ which is $$(2x-3)(2x-3)$$ Then I divide the first fraction denominator by my ...
2
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2answers
56 views

Mathematics language, how to say that a specific value of $x$ is included in the functions domain?

For instance, we have the function $y=-2(x+1)^2-10$. $x=0$ is included in this functions domain. How can I say this mathematically, instead of typing out a sentence and saying that $x=0$ exists in ...
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1answer
178 views

How many points do the graphs of the following functions have on the $x$-$y$ axis? Infinite or finite?

I am stuck on this question and cannot figure it out, $$y=-4-x$$ $$y=\frac{1}{x}-x$$ The first equation is a line, so should it not have infinite points? The second equation has a restriction, $x ...
2
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2answers
30 views

Exponents with Logs

Could someone show work for why $e^{2\ln(x)}$ = $x^2$ ? I ran across this while solving an ODE but have completely forgotten the rules used here. I hate to ask it, but i'd rather ask it this once than ...
0
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2answers
117 views

Prove symmetry of natural logarithm

Prove that $f(x)=\ln\sqrt{x^2+1}$ is symmetrical in $x=0$. $\ln\sqrt{(x-a)^2+1}=\ln\sqrt{(x+a)^2+1}$ $\sqrt{(x-a)^2+1}=\sqrt{(x+a)^2+1}$ $(x-a)^2+1=(x+a)^2+1$ $x^2-2ax+a^2+1=x^2+2ax+a^2+1$ ...
0
votes
1answer
62 views

$\frac{x}{y} \ge \frac{a_1}{b_1} \ge \frac{a_2}{b_2}$ and $b_1 \le b_2 \implies \frac{x+a_1}{y + b_1} \ge \frac{x+a_2}{y + b_2}$?

Given $\frac{x}{y} \ge \frac{a_1}{b_1} \ge \frac{a_2}{b_2}$, where $x,y,a_i,b_i$ are positive numbers. I would like to prove the following: Claim: If $b_1 \le b_2$, then $\frac{x+a_1}{y + b_1} ...
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1answer
44 views

Draw graph of $y=x^2$ using these equations

Draw graph of $y=x^2$ using these equations: $x^2-4x+3=0$ $x^2-7=0$ $x^2-2x+5=0$ I don't understand how to put these equations in a form from which they are ready to be plotted on a graph. Do I ...
0
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3answers
55 views

calculate the derivative of $x + 1/x$ using the definition?

Calculate the derivative of $x + 1/x$ directly from the definition of the derivative $$ \lim_{h\to0}\frac{f(x+h)-f(x)}{h} $$ I think this is the first step: $((x+h) + 1/(x+h) -(x+1/x) )/h$? but I'm ...
0
votes
1answer
91 views

On the existence/applications of infinitely-nested functions

Inside a previous question, one particular nested function shown is the known tetration. This "kind" of arbitrary repeated functions has always intrigued me, because inside their properties lie so ...
2
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1answer
72 views

procedure of proving that a number is rational [closed]

How can I prove the following ? :
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2answers
34 views

Help with Evaluating a Function

Find the difference quotient $$\frac { f(a+h)-f(a) }{ h } ,\quad where\quad h\neq 0$$ for the function: $$f(x)=\frac { x }{ x+1 }$$ Steps that I took to try and solve this: $$\frac { \frac { a+h }{ ...
0
votes
2answers
81 views

Solve $yx^2-zx+v=0$ for x?

I am having trouble solving $yx^2-zx+v=0$. $y$,$z$, and $v$ are constants. I cannot just plug into the quadratic formula whilst solving this. I'll post what I have done even though I do not think I am ...
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4answers
172 views

Factoring the following polynomials

Factorize the following polynomial: $t^3 -9t +8$ $t^6 -91t^2 +90$
2
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3answers
326 views

Simplifying the sum of powers of the golden ratio

I seem to have forgotten some fundamental algebra. I know that: $(\frac{1+\sqrt{5}}{2})^{k-2} + (\frac{1+\sqrt{5}}{2})^{k-1} = (\frac{1+\sqrt{5}}{2})^{k}$ But I don't remember how to show it ...
0
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2answers
563 views

Every non-empty subset of the integers which is bounded above has a largest element.

I was reading a proof about every non-empty subset of the integers which is bounded above has a largest element, but i have troubles in one step. Here is the proof: Since $S$ is a non-empty subset of ...
1
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1answer
33 views

Rate of change vs. average rate of change of a function

What is the difference between the both? I am trying to find a separate definitions for both in a textbook, but it only defines average rate of change. I know that average rate of change is: $$\frac ...
0
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2answers
34 views

Find angle $\alpha$ from a complex vector

I'm trying to solve this problem from a Russian book: Find the angle which is needed to rotate the vector $3\sqrt{2} + i2\sqrt{2}$ to obtain the vector $-5+i$. EDIT: $\tan\dfrac{\pi}{6} \neq ...
4
votes
6answers
292 views

When $n$ is divided by $14$, the remainder is $10$. What is the remainder when $n$ is divided by $7$?

I need to explain this to someone who hasn't taken a math course for 5 years. She is good with her algebra. This was my attempt: Here's how this question works. To motivate what I'll be doing, ...
1
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2answers
208 views

Calculate $n$ points having equal cartesian distance over a single sine wave

I'd like some help figuring out how to calculate $n$ points of the form $(x,\sin(x))$ for $x\in[0,2\pi)$, such that the Cartesian distance between them (the distance between each pair of points if you ...
3
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2answers
133 views

Simplify rational expression

How do I simplfy this expression? $$\dfrac{\frac{x}{2}+\frac{y}{3}}{6x+4y}$$ I tried to use the following rule $\dfrac{\frac{a}{b}}{\frac{c}{d}}=\frac{a}{b}\cdot \frac{d}{c}$ But I did not get the ...
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2answers
128 views

Proof of period of $f(ax+b)$

I have been taught that $f(x)$ is called a periodic function with period $T$ if $$f(x)=f(x+T)$$ This I understand completely. Also I have been taught that $$f(ax)=f(ax+T/|a|)$$ if $f(x)$ has a period ...
1
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1answer
73 views

Modifying expression with exponent to another form

I did a math exercice where I had to prove that an expression was equal to the expression on the left hand side, so I took my expression, transformed it, and eventually I came up with the expression ...
4
votes
2answers
3k views

Solving a logarithmic expression without a calculator

How do I find the value of this logarithmic expression without using a calculator? I'm trying to relearn algebra, but this problem has me scratching my head, and my Google tutorial searches are ...
0
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1answer
33 views

How to simplify a quotient which has the difference of squares in the numerator?

How do I simplify this expression? $$\frac{10x^2-0.1a^2}{x-0.1a}$$ I tried to expand the expression $$\frac{10(x^2-0.01a^2)}{x-0.1a}$$ but don't know where to go from here.
0
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1answer
40 views

Sketching the curve $\frac{y^2}{a^2}+1=\frac{a}{x}$

Firstly, if don't understand the domain of the function as the constant $a$ is not specified to be positive or negative, (correct me if the sign doesn't make a difference). I tried finding the ...
0
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1answer
190 views

If $\tan^2\theta=1-e^2$, then the value of $\sec\theta+\tan^3\theta \csc \theta$ is

If $\tan^2\theta = 1-e^2$,then the value of $\sec\theta$ + $\tan^3\theta \cdot \csc\theta$ is... NOTE: $1/\cos\theta +\cos^3\theta/\sin^3\theta \cdot 1/\sin\theta$ multiply: $\sin^2 ...
2
votes
1answer
49 views

About the definition of complex multiplication

Some people say that the complex product is the way it is to respect the distributive law of multiplication. However, the distributive law acts in the whole number, like: $$(a+b)(c+d) = ac + ad + bc ...
0
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1answer
78 views

graphing hyperbola algebra problem

I have the hyperbola from a textbook 9x^2 - 18x - 16y^2 - 64y = 91 It is supposed to become: ((x-1)^2) / 4 - ((y+2)^2) / (9/4) = 1 I cannot get this though, I arrive at: ((x-1)^2) / 4 - ((y+2)^2) / ...
0
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1answer
27 views

Finite points from graph of the following equations?

The graphs of the following equations on the x-y axes have how many points: no points, a finite number, or infinitely many? $x^2 + y^2 = -4$. $x+y=-4$, $x^2+xy=1$ Don't all of these equations have ...
5
votes
3answers
170 views

Determining whether points are collinear

$(1,1)(3,9)(6,21)$ The way I figured that this should be solved is by finding the slope of: $(1,1)(3,9)$ Then, $(3,9)(6,21)$ Finally $(1,1)(6,21)$ Which are 4, 4,and 4 respectively. So I ...
4
votes
2answers
87 views

Slope of a vertical line

Find the equation of the line that satisfies the given conditions: Through $(-1,2)$; parallel to the line $x=5$ I know that the equation of this line is $x=-1$ because $x=5$ is a vertical line and ...
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1answer
27 views

What would the answer be using Linear Equation

How many ways to arrange HATE...............................................
1
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2answers
34 views

Help solve this Problem

Given that $a +b +c =5$ and $ab +ac +bc=$5, what is the value of $a^2+b^2+c^2$?
2
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3answers
669 views

Find the length of a ladder that is leaning against a wall.

A ladder leans against a wall at a point $8$ feet above the ground. The bottom of the ladder rests $2$ feet away from the wall. How long is the ladder?
0
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1answer
155 views

Graphing an inequality to solve a mathematical model

An appliance manufacturer estimates that the profit $y$ (in dollars) generated by producing $x$ cooktops per month is given by the equation $$y=10x+0.5x^{ 2 }-0.001x^{ 3 }-5000$$ where $0\le x\le ...