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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0answers
39 views

Solving a SAT algebra question

I have the equation: $$a(x+1) + a - b = 2$$ Find the value of a: a) $0$ b) $1$ c) $2$ d) $3$ This question is from a SAT Math 2 practice test. I have no idea what to do. Help is appreciated. ...
4
votes
3answers
364 views

Approximation of the Sine function near $0$

What is the reason that for $x<0.5$, $\sin(x)\approx x$? Are there more known properties of these kind for other trigonometry functions?
2
votes
3answers
49 views

How to define $y= |x+2|+|x-3|$ in a piecewise manner

I need to define the function $y= |x+2|+|x-3|$ over the relevant intervals, but I am not entirely sure what this entails. How do I find the needed intervals? Plugging in different values gives me an ...
0
votes
3answers
49 views

Precalculus Roots of Unity

Let $A$ be the set of all complex numbers $z$ such that $z^{24}=1$ and let $B$ be the set of all complex numbers $w$ with $w^{54}=1.$ That is \begin{align*} A&=\{z\;|\;z^{24}=1\}\\ ...
0
votes
0answers
6 views

Coordinate - Resizing and Positioning

I have an image that is 1000x134 referred to as logo I have another view in the back The user is allowed to pan, pinch, and rotate the logo to position it where ever they want on top of the ...
5
votes
4answers
840 views

Albert, Bernard and Cheryl popular question (Please comment on my theory)

Here is the problem, I think that there is one point that makes the question ambiguous, I think they should explicitly say the reason why Albert knows that Bernard does not know the date. Case 1: ...
1
vote
2answers
64 views

Help with double simultaneous equations and roots

I am in the course of a project, in which I need to solve these two simultaneous equations: \begin{equation} \sqrt{(1000-y)^2 + x^2} - \sqrt{y^2 + x^2} = 342.371 \end{equation} \begin{equation} ...
1
vote
2answers
40 views

How can I reduce the occurrence of algebra mistakes in my solutions?

Apologies is this is the wrong forum; point me to the right one if that's the case. In this particular case, I'm studying limits (in an intro Calc course, and on Khan Academy) but the question ...
0
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1answer
23 views

Using Partial Fraction Decomposition to acquire appropriate form for GCIF

I need to find the PFD so I may continue with a complex integral $\int_C \frac{ze^z}{z^6 - 1}dz$, $z \in \mathbb{C}$. The contour $C = |z-a|=a$, $a>0$ I have found all $6$ roots of $z^6 - 1$, so ...
1
vote
0answers
51 views

Evaluate the following series using elementary methods. [duplicate]

$\frac{1} {\sqrt 1} + \frac{1} {\sqrt2} + \dots + \frac{1} {\sqrt{ 100}}$ Just bear in mind that I'm going to say the solution to a person of low education. So please provide creative hints or ...
-1
votes
0answers
21 views

Number of escalator steps we can see [closed]

A man walks up an escalator that moves up and counts 50 steps. The next day he walks up the same escalator and counts 75 steps. If the second speed (in steps per time unit) is three times the first ...
0
votes
4answers
40 views

Illegitmate inference in solving a linear equation (which one?)

Consider the following five ways of solving the equation $3(r - 5) = 24$: (1): $$3(r - 5) = 24$$ $$\frac{3}{3}(r - 5) = \frac{24}{3}$$ $$r - 5 = 8$$ $$r = 8 + 5$$ $$r = 13$$ (2): $$3(r - 5) = ...
-1
votes
2answers
37 views

$y=3^{\cos(x)}$ how to graph this goniometric function

Please help me with graphing this function $y=3^{\cos(x)}$ without grapher. Thanks in advance for all your procedures.
1
vote
2answers
51 views

How to find value of $x$ in this formula

I have this formula: $$1-\frac 1x=y$$ How do I invert this so that, if I have value of $y$, I want to find value of $x$. I know, but I am pretty dense in math :( I dont even know what category to ...
3
votes
3answers
555 views

Question about a solution of a system of three non linear equations in three unknowns

Let $a$, $b$ and $c$ be positive real numbers such that $$ a + \frac{1}{b} = 3$$ $$b + \frac{1}{c} = 4$$ $$ c + \frac{1}{a} = \frac{9}{11} $$ then $$ a \times b \times c =?$$ I tried doing this ...
0
votes
1answer
47 views

Convert non-linear equation into linear form i.e. (Y=mX+c)

How would you go about converting ${1\over y} = x^2 + a^{b-x} $ into linear form. I know how you would normally go about solving this type of problem but I fail to make any progress on this one. I ...
0
votes
1answer
28 views
1
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3answers
41 views

System of equations with radicals

Solve the system of equations (in $\mathbb R$): $$\begin{matrix} 2\sqrt[4]{\frac{x^4}{3}+4}=1+\sqrt{\frac{3}{2}y^2} \\ 2\sqrt[4]{\frac{y^4}{3}+4} = 1+\sqrt{\frac{3}{2}x^2} \end{matrix}.$$ This ...
1
vote
2answers
35 views

Equation discrepancy ?

Here's a (possibly dumb) question I found If sqrt(2x-1) = -x , find x The way to solve it is essentially ...
0
votes
1answer
17 views

Regarding least squares, value of n in a scatter plot

I am currently in a college algebra class wherein I am required to do a rather lengthy project regarding least squares. One particular exercise posits the following (keep in mind that in this project ...
2
votes
2answers
40 views

Roots of Unity for Precalculus

(a): What is the smallest positive integer $n$ such that all the roots of $z^4 + z^2 + 1 = 0$ are $n^{\text{th}}$ roots of unity? (b) What is the smallest positive integer $n$ such that all the roots ...
10
votes
1answer
105 views

Identity in Ramanujan style

Is it possible to represent $$ \sqrt[3] {7\sqrt[3]{20}-1} =\sqrt[3]{A}+\sqrt[3]{B}+\sqrt[3]{C}$$ with rational $A,\,B,$ and $C?$
3
votes
2answers
56 views

Simplify $ \frac{b}{a-b}+ \frac{a}{b-a}$

I have a question regarding simplification of an algebraic expression. Here is the problem: $ \frac{b}{ a - b} + \frac{a}{b-a} $ The outcome is $ -1 $ Here is how I try to simplify it: Add ...
0
votes
1answer
22 views

Number of integral values of M

If $\log_3M=a_1+b_1$ and $\log_5 M=a_2+b_2$, where $a_1,a_2$ are natural numbers and $b_1,b_2 \in [0,1)$. If $a_1a_2=6$, then find the number of integral values of M. What so I do in the problem. I ...
3
votes
1answer
41 views

System of logarithmic equations

$$\log (2000xy)-\log x\log y=4$$$$\log(2yz)-\log y\log z=1$$$$\log(zx)-\log z\log x=0$$ The base is 10 everywhere. I tried opening the log with the sum formulae and then manipulating, but I got stuck. ...
0
votes
2answers
55 views

How do $\ln$ and $e$ cancel out when there is an exponent that is negative?

I have a question about $\ln$ and $e$. Essentially I know that $e$ and $\ln$ cancel/reverse each other to give $x$. Ex: $e^{\ln x} = x$ But how does this work: $-e^{-\ln 2} = -1/2$ I thought it ...
0
votes
2answers
42 views

Factoring quadratic expression $25t^2-16r^2$

When I factor this quadratic $25t^2-16r^2$ expression I get the following: $$25t^2-16r^2 = 25t^2 + 20t - 20t + 16r^2 = 5t(5t-4) + 4(5t-4r^2)$$ But the $(5t-4)$ and $(5t-4r^2)$ aren't identical, ...
-1
votes
2answers
68 views

Proving by induction $2^k - 1 = 1+\cdots +2^{k-1}$

How can I show: $$2^k - 1 + 2^{(k+1)-1} = 2^{k+1} - 1$$ I am trying to prove this by induction: $$2^k - 1 = 1+\cdots +2^{k-1}$$ and proved the base case: $2^2-1 = 1+2^1$ as $2^2-1=3$ and ...
-1
votes
1answer
55 views

How can I disprove that? [duplicate]

Prove that 5=-5 $$ \sqrt{(-5)^2} = \sqrt{25} = 5 = \sqrt{(-5)^2} = \sqrt{(-5)\cdot(-5)} = \sqrt{(-5)} \cdot \sqrt{(-5)} = (i \sqrt{5})\cdot(i\sqrt{5}) = -5\,. $$
0
votes
1answer
15 views

Algebra manipulation for a integral

When deriving the integral ${\int_0^3(x^3-6x) dx}$ in terms of reimann sums it has these two steps in my calculus book, Stewart 7th edition, and I don't understand how to derive the 2nd from the ...
0
votes
1answer
25 views

How to find the domain of the function $f(x) =\log_y a^2$?

To find the domain of the function $f(x) = \log_y a$ it's enough to check if the base (y) is greater than 0 and not equals to 1 and the number (a) is greater than 0. But what if we have a power of ...
0
votes
0answers
27 views

Order of subtraction of an expression in algebra

Can there be more than one 'answer' in computing an algebraic expression? Case in point, consider $$ \frac{6ab}{2} - \frac{ab}{2} - \frac{2ab}{2} $$ If I compute the terms from left to right, I get ...
1
vote
3answers
78 views

Solve $\frac{(x-1)^{204}(x+3)^5(x-4)^{2015}}{(x+5)^{102}}\ge 0$

Solve $\frac{(x-1)^{204}(x+3)^5(x-4)^{2015}}{(x+5)^{102}}\ge 0$ Just wanted to share a nice and quick technique i learnt for such problems.
0
votes
3answers
72 views

Solve $\frac{|x|}{|x-1|}+|x|=\frac{x^2}{|x-1|}$

Solve $\frac{|x|}{|x-1|}+|x|=\frac{x^2}{|x-1|}$.What will be the easiest techique to solve this sum ? Just wanted to share a special type of equation and the fastest way to solve it.I am not asking ...
0
votes
3answers
139 views

Solving equation containing different terms of the form x^x

Is it possible to solve the following equation for $x$ as a function of $y$: $$\sqrt{\frac{x+k}{x}}\,\frac{(x+k)^{x+k}}{x^x}=y$$ in a way that the resulting equation $x=f(y)$ is something I can ...
1
vote
4answers
73 views

If $16^{\sin ^2x}=5$, then what is $2^{\cos^2x}$?

I happened to create this problem and solved it. I used only basic algebra and trigonometry. I thought it was a fun problem, so I wanted to expose the problem to the public. Please provide an exact ...
0
votes
1answer
33 views

Factoring and solving a cubic polynomial

When can we not use synthetic division to solve for a cubic polynomial? For example we can use synthetic division to solve $-t^3 -4t^2 +20t +48$. When I can't use synthetic division what are my other ...
2
votes
1answer
56 views

How many asymptotes does $y=\frac{x^2}{x-2}$ have?

In a question I came upon, the answer insisted that there were three; one was apparently a horizontal asymptote, which I do not agree with. There are only 2 asymptotes, correct? One is $y=x+2$ and the ...
0
votes
1answer
33 views

Solving for single variable proving to be extremely difficult.

I have been at this equation for about two days now, and I can not for the life of me find a way to solve to i. If anyone can please show me a step by step into solving this, it would help me out so ...
0
votes
1answer
33 views

Domain of $\left(f(x)\right)^a$ where $a$ is an irrational number.

Why, if $f(x)$ is a real function and $$\left(f(x)\right)^a$$ where $a$ is an irrational number, we put $$f(x)>0$$ for its domain?
3
votes
3answers
43 views

Forming Partial Fractions

Suppose we have: $ \frac{f(x)}{g(x)h(x)} $ and we want to break it down into; $ \frac{I(x)}{g(x)} + \frac{J(x)}{h(x)}$ and that; $deg(f) \leq deg(g)+deg(h)$ , $deg(i) < deg(g)$, $deg(j) ...
2
votes
0answers
28 views

Polynomials and Divisibility Rule.

The question is this - If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $h(x)=xf(x^3)+x^2g(x^6)$ is divisible by $x^2+x+1$, then which of the following are true? 1. $f(1)=g(1)$ ...
1
vote
2answers
40 views

Algebra and solving for n

$$162\left(1-\left(\frac{1}{3}\right)^n\right) -162=-0.05$$ Solve for n I've tried myself but am getting 2.something and the answer should be 7.36. I know you need to use logs but not working for me ...
2
votes
1answer
59 views

What did I do wrong trying to find this limit?

In another question, a user asked to find: $$\lim_{x\to 0} \frac{\exp(x^2)-\cos(x)}{\sin(x)^2}$$ I thought I could use pure trigonometric identities to find the limit. Apparently I was mistaken, but I ...
3
votes
1answer
56 views

Find integer $n$ that satisfies $(\lg n)^{2^{100}} <\sqrt{n}$ with $n > 2$

If $(\lg n)^{2^{100}} < {n^{1/2}}$, where $\lg$ is the binary logarithm, then $$(\lg n)^{2^{101}} < n$$ $$2^{101}\lg \lg n < \lg n$$ $$101 < \lg \lg n - \lg \lg \lg n$$ I don't know that ...
0
votes
2answers
60 views

How can I find the derivative of this integral?

A function is defined for a constant $x$ after integrating out with variable $t$ as: $$ F(x) = \int_0^4 \log(1-x^2t^2)\,dt $$ making it as a function of $x$. How can I now find $ F(x=0)$ and $ ...
1
vote
0answers
37 views

Is there a companion to the book 'A Synopsis of Elementary Results in Pure and Applied Mathematics' by George S. Carr?

A Synopsis of Elementary Results in Pure and Applied Mathematics by George S. Carr is as most of you probably know a book that was famously used by the great mathematician Ramanujan. It is said he ...
6
votes
8answers
107 views

How to show that $6^n$ always ends with a $6$ when $n\geq 1$ and $n\in\mathbb{N}$

Is there a proof that for where $n$ is a natural number $$6^n$$ will end with a $6$? I can understand conceptually that $6\cdot 6$ ends with $6$ and then multiplying that by $6$ will still end with ...
-1
votes
2answers
52 views

Finding a single irrational root to a rational function.

The function is: $$y={x^3 + 3x^2 + 6\over x-3}$$ I have to do a sketch and was able to find $y$-int, vertical asymptote, end behavior asymptote. When searching for roots I used rational root ...
1
vote
1answer
47 views

numerical question from practice test

If I may could I ask for help on the following question taken from SHL practice test? Jason is considering purchasing a new machine to make plastic silverware. The machine produces 1,000 pieces of ...