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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2
votes
2answers
71 views

Knowing when to use factoring/how to factor/rational zeroes

How do I know where I would be using factoring as opposed to rational zero theorem? Do I do Descartes rule of signs to get how many positive/negative and then attempt RZT to get rational zeroes, then ...
4
votes
3answers
233 views

Solving this equation $10\sin^2θ−4\sinθ−5=0$ for $0 ≤ θ<360°$

The first part of the question asks me to square both sides of the equation: $$3 \cos θ=2 − \sin θ$$ So that I can obtain and solve the quadratic: $$10\sin^2θ−4\sinθ−5=0 \;\;\text{for}\;\; 0 ≤ ...
2
votes
4answers
796 views

Explain why there's no solution to the equation $2x-2x^2 = 1$

How can you tell there's no solution to the equation $2x - 2x^2 = 1$. The supporting information goes like this: The diagram above shows the graph of $y = 7 + 2x - 2x^2$. I tried to do this: $2 ...
6
votes
5answers
390 views

Simplify $ \frac{1}{x-y}+\frac{1}{x+y}+\frac{2x}{x^2+y^2}+\frac{4x^3}{x^4+y^4}+\frac{8x^7}{x^8+y^8}+\frac{16x^{15}}{x^{16}+y^{16}} $

Please help me find the sum $$ \frac{1}{x-y}+\frac{1}{x+y}+\frac{2x}{x^2+y^2}+\frac{4x^3}{x^4+y^4}+\frac{8x^7}{x^8+y^8}+\frac{16x^{15}}{x^{16}+y^{16}} $$
1
vote
2answers
165 views

Show $ \frac{x}{xy+x+1}+\frac{y}{yz+y+1}+\frac{z}{zx+z+1}=1 $ given $xyz = 1$

Please help me prove the equality: If $xyz=1$, prove that $$ \frac{x}{xy+x+1}+\frac{y}{yz+y+1}+\frac{z}{zx+z+1}=1 $$
3
votes
4answers
121 views

How to approach solving this indices example?

I am having problems working out how to approach solving this problem : $$\left(\frac{81}{16}\right)^n = \frac{32}{243}$$ How do I go about working out $^n$? Please step by step if possible.
3
votes
3answers
58 views

How to get from $1 + (-1)^{n+1}$ to $1 + [((−1)^{n }) − 1] (−1)$

I need some help with the algebra here. I have the following explanation, and I really can't follow the algebra. Could you also maybe give me some tips on how to think about such problems. $a_n = 1 ...
8
votes
2answers
112 views

How to solve $4^x - 2^{x + 1} = 3 $ for x?

We figured that this can be changed to $2^{2x} - 2^x \cdot 2 = 3$, but couldn't solve from there. Perhaps we are not on the right path?
3
votes
3answers
1k views

How do I find the maximum and minimum values of $1−4\cos(2\theta)+3\sin(2\theta)$?

To find the maximum of $$1 - 4\cos(2\theta) + 3\sin(2\theta) $$I tried: $$1-4(1)+3(1)=0.$$ To find the minimum I tried to substitute with the minimum values of sin and cos: $$1-4(-1)+3(-1)=2$$ I know ...
0
votes
3answers
152 views

Is this equation solvable for x?

Is is possible to solve this equation for x? $$ \frac{x}{q} + \frac{x}{(1-xk)^2} - t = 0 $$ x, q, k, t are all Real and positive.
3
votes
2answers
75 views

How to simplify square root

$$ \sqrt[3]{a}(\sqrt[3]{a^2}-\sqrt[3]{a^5}) $$ How can this be simplified? I can't find anything for doing the subtraction.
1
vote
3answers
443 views

Show that this function is one to one

I am trying to show that this function $$s:R−>R$$ defined by $$s(x)=(e^x - e^-x)$$ is one to one. My approach is using the natural $\log{} (\ln{})$ to cancel out the $e$'s: $e^x - e^{-x} = e^y - ...
2
votes
3answers
71 views

How to differentiate $F(y) = \left(\frac{1}{y^2}-(\frac{-2}{y^4})\right)\cdot\left(y+5y^{3}\right)$

I tried to differentiate this function, and was able to differentiate the second portion $(y + 5y^3)$, but I could not differentiate the $\left(\frac{1}{y^2}-(\frac{-2}{y^4})\right)$ term. Now I know ...
4
votes
1answer
116 views

Is Algebra Closed under all algebraic operations?

Note the following: If you take the set of integers $\mathbb Z$: and the operations of $+$ and $-$ Then all equations of the form ($x + a_1 + a_2 + a_3+\cdots+ a_N = b$) where $a$'s and $b$ are ...
1
vote
1answer
179 views

Prove that $\cos2\theta−\sqrt{3}\sin2\theta \equiv2 \cos (2\theta+\pi/3 )\equiv−2\sin(2\theta−\pi/6)$

Going from $2\cos(2\theta+\pi/3)$ to $\cos2\theta−\sqrt{3}\sin2\theta$ is simple enough, however I'm stuck on going from $2\cos(2\theta+\pi/3)$ to $−2\sin(2\theta−\pi /6)$. How do i do this?
2
votes
4answers
238 views

Solve an equation containing ceil function

I have a pretty simple formula: $y=x+7\lceil\frac x {1113}\rceil$, but I don't know how to solve it for $x$. For example, if $x$ is $5520$, $y$ would be $5520+7\lceil\frac {5520} {1113}\rceil=5555$. ...
0
votes
3answers
82 views

Circle- basic question [closed]

I know this is a basic question, but if I have a circle with radius 2 and I look at the area on the circle between angles $-\pi$ and $\pi$, will that be the bottom half of the circle?
1
vote
1answer
54 views

Graphing equations

I am learning how to graph equations. I think $y=x$ is a line and $y=x^2$ is a parabola. I have a wiggly graph. The question is to find an equation for the graph. It then asks given any graph can you ...
0
votes
1answer
365 views

Calculate percentage on negative to positive scale [closed]

Scale 1: -5 -4 -3 -2 -1 0 1 2 3 4 5 Scale 2: 1 2 3 4 5 6 7 8 9 10 11 How do I calculate the percentage of ...
2
votes
3answers
77 views

Find at least two ways to find $a, b$ and $c$ in the parabola equation

I've been fighting with this problem for some hours now, and i decided to ask the clever people on this website. The parabola with the equation $y=ax^2+bx+c$ goes through the points $P, Q$ and $R$. ...
0
votes
1answer
16 views

solutions and graph

For this graph, we are graphing all of the time this person is walking or resting. If he takes a break to rest for 10 mins, the graph will not stop. It should continue. Time will continue, but he will ...
0
votes
1answer
22 views

Algebra for derivative set to greater than zero

I have this equation $\frac{db_2}{dt} = q_2b_2(1-b_2-(1 - \frac{a}{q_1}))- q_1b_2(1 - \frac{a}{q_1}) - ab_2$, where $b2$ is negligibly small but greater than zero (that is I intend on treating as a ...
7
votes
1answer
462 views

factorise, $x^3-13x^2+32x+20$

factorise, $x^3-13x^2+32x+20$ Let, $f(x)=x^3-13x^2+32x+20$ $f(x)=x(x^2-13x+30)+2x+20$ $f(x)=x(x-3)(x-10)+2x+20$ $f(-1)\lt 0$, $f(0)\gt 0$, which shows there is a root between $x=-1$ and $x=0$ ...
1
vote
2answers
49 views

Combination and product question

There are $5$ numbers, and each combination of $4$ numbers from those $5$ has a product of either $10, 20, 30, 40$, or $50$. What is the quotient of the sum of those $5$ numbers divided by the product ...
9
votes
7answers
1k views

How to simplify a square root

How can the following: $$ \sqrt{27-10\sqrt{2}} $$ Be simplified to: $$ 5 - \sqrt{2} $$ Thanks
1
vote
1answer
101 views

How to do this indices example?

Beginner question, I have been trying to figure out how my book got the solution it did for this question : $$\frac{(2r^2)^5 (3r^4)^3}{(6r^3)^2} = 24r^{16} $$ I get the $r^{16}$ part but how did ...
1
vote
0answers
258 views

Unit-circle versus trigonometric approach to introducing Calculus

I'm trying to decide on a text for a refresher in pre-Calculus. The publisher of two books by the same set of authors differentiates between them as follows: Precalculus, Seventh Edition This ...
0
votes
1answer
250 views

Converting $5\log$ reduction to a percentage

I see a lot of terms bounded about like a 5log reduction of bacteria. So I presume it's $5\log_{10}$. Now how do I convert that into a percentage? Would it be: $$\begin{align} ...
1
vote
2answers
62 views

How to find $a,b\in\mathbb{N}$ such that $c = \frac{(a+b)(a+b+1)}{2} + b$ for a given $c\in\mathbb{N}$

Suppsoe that $$c = \frac{(a+b)(a+b+1)}{2} + b$$ Now $c$ is given - how does one find satisfying $a, b$?
2
votes
2answers
6k views

What is the square of summation?

Consider the following, which one of the following is true ?? $$\left( \sum^{n-1}_{j=0}Z_j\right)^2 = \sum^{n-1}_{j=0} Z_j^2 + \sum^{n-1}_{j\neq i} Z_i Z_j$$ OR $$\left( ...
2
votes
2answers
112 views

question on transformation

If a $2$d coordinate transformation function is given by $f(x,y)= 3x+1$, then what does it mean? How do I calculate the transformed coordinates for the points say $(3,4)$ in the initial space?
0
votes
2answers
28 views

For what range does this floor function scale to?

I have $\lfloor\frac{X}{(2y+1)^2}\rfloor = k$ where $X$ and $k$ are known. For what values of $y$ will this hold true? edit: all are positive integers
1
vote
3answers
248 views

Find a formula that generates the following sequence

Find a formula which generates the following sequence. $$15,20,25,30,35 \ldots $$ The answer is $5(n + 2) $ How? I know it comes from the formula $a_n = a_1 + (n - 1) d$, but I am not sure how they ...
2
votes
1answer
159 views

A Question About Linear Interpolation

So lets say I have two points $A=(x_1, y_1, z_1)$ and $B=(x_2, y_2, z_2)$. $A$ and $B$ are each associated with some scalar value $K_1$ and $K_2$. $K_1$ is negative and $K_2$ is positive and all the ...
1
vote
1answer
72 views

Find the equation of the hyperbola.

Given: vertical asymptote = $\frac{5}{4}$ Passes through points: (1,5) and $\left(\frac{-5}{2},\frac{11}{5}\right)$. Because we are given vertical asymptote , we should have a rectangular hyperbola: ...
4
votes
3answers
141 views

Order of calculation in all math equations

I already asked a question (Order of operations in rotation matrix notation.) about the order in which a particular equation is "processed" and now I need to generalise that and learn the rules of ...
-1
votes
2answers
78 views

point belong to all tangents of $xe^x$.

Given a function $f$ defined on $\mathbb{R}$ as : $f(x)=xe^{-x}$, it graph is $(C)$. Question : are there any point belong to all the tangents of $(C)$ ? This is a part from a generalized problem ...
2
votes
3answers
63 views

Range of $\frac{1}{2\cos x-1}$

How can we find the range of $$f(x) =\frac{1}{2\cos x-1}$$ Since range of $\cos x$ can be given as : $-1 \leq \cos x \leq 1$ therefore we can proceed as :$$\begin{array}{rcl} -2 \leq & ...
5
votes
2answers
111 views

$ \lim_{n\to\infty}{\left(\frac12\cdot\frac34\cdot\frac56\cdots\frac{2n-1}{2n}\right)}=0 $

Prove that $$ \lim_{n\to\infty}{\left(\frac12\cdot\frac34\cdot\frac56\cdot\ldots\cdot\frac{2n-1}{2n}\right)}=0. $$ Transforming it to factorial obviously doesn't help at all, so I've noted ...
1
vote
1answer
51 views

If a bacterium multiplies for 4 every 1 minute, in 6 mins how bacteria there will be?

If a bacterium multiplies for 4 every 1 minute, in 6 mins how bacteria there will be? Please, could you answer this with an explanation, or the calculation you used to get to the result.
1
vote
2answers
89 views

Evaluate $(-3)^2$, $3^2$ and $-3^2$.

Evaluate each of the numeric expressions $(-3)^2$, $3^2$ and $-3^2$. When I originally asked this problem I confused it with another problem...please help...thank you
1
vote
2answers
67 views

Evaluate each of the numeric expressions: $\sqrt{(-3)}$, $\sqrt{3}$, $\sqrt{-3}$

Need help breaking down and understanding the concept to get answer.
2
votes
1answer
56 views

Advance math Stem Homework for Middle School Students in the sixth grade.

I am helping out my sixth grader, great niece with her math problem. Determine the value of $(-1/2)^2$. Please help.
6
votes
3answers
355 views

When does $x_1+x_2+\dots+x_n=a,\space x_1^2+x_2^2+\dots+x_n^2=b$ have a unique solution?

Let $n\in \Bbb N$ be fixed. For which $a,b\in \Bbb R$ does the equation system $$x_1+x_2+\dots+x_n=a$$ $$x_1^2+x_2^2+\dots+x_n^2=b$$ have a unique solution for the $x_i$? For example if ...
1
vote
1answer
64 views

$z= \frac{u-\overline{u}v}{1-v}$ is real is equivalent to $|v|=1$.

Let $u,v$ be complex numbers such as $u,v\notin \mathbb{R} $, and : $$z= \frac{u-\overline{u}v}{1-v}$$ Prove that : $z\in\mathbb{R} \Longleftrightarrow |v|=1$.
4
votes
3answers
239 views

Weird math question in ACT prep

I do not even know where to begin here. I was taking one of the ACT prep quizzes for school the other day and came across a question that I cannot even begin to understand. For all $a$ and $b$ such ...
3
votes
2answers
62 views

Solving: $\frac{3x-1}{2} =\frac{-2}{x+2}$

How to solve : $$\frac{3x-1}{2} =\frac{-2}{x+2} $$
1
vote
1answer
86 views

Localization of a ring which is not a domain

Let $A$ be a ring (commutative with $1$), let $S$ be a multiplicatively closed subset of $A$, i.e $S$ is contained in $A$ , $1\in S$ and $a,b\in S$ implies $ab\in S$, for every $a,b\in A$. Consider ...
8
votes
3answers
858 views

Showing that $\max\{f+g\} \leq \max f + \max g$

Given real-valued continuous functions $f, g$, is the following (and why?) inequality true? $$\max \{f + g \} \leq \max f + \max g$$ Can someone give me a proof? I suspect the min is the reverse ...
1
vote
1answer
86 views

help in a continuity problem of piecewise function

if $f(x)=x^2-2|x|$ , then we have to test the differentiability of $g(x)$ in the interval $[-2,3] $ , where $$g(x) = \begin{cases} \min\{f(t); -2≤t≤x\}&: x \in [-2,0)\\ ...