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
0
votes
1answer
21 views
Relationship between 2 Dimensional Quadratic systems and roots
Given four points
$(x_1, y_1)
(x_2, y_2)
(x_3, y_3)
(x_4, y_4)$
How does one construct a system of two equations:
$a_1x + a_2x^2 + a_3y + a_4y^2 + a_5xy = c_1$
$b_1x + b_2x^2 + b_3y + b_4y^2 + ...
1
vote
1answer
57 views
if you invest $ 500 at 6 % compounded annually,
Please help me with this problem. It needs to be done in the same format as below
if you invest $500$ dollars at $6$ percent compounded annually, how many years to the nearest tenth would it take your ...
2
votes
3answers
82 views
Solving two algebraic equations
I'm trying to prove an identity from physics. I have the following two equations ($M$ is a constant):
$$e^2 = \left(1-\frac{2M}{r}\right)\left(1+\frac{l^2}{r^2}\right)$$ and $$r = ...
3
votes
2answers
39 views
$f(g(x))=x$ implies $f(x)=g^{-1}(x)$
Is it possible to find a necessary and sufficient condition to conclude when
$$f(g(x))=x \implies f(x)=g^{-1}(x) \wedge f^{-1}(x)=g(x),$$
if both functions are well defined?
3
votes
4answers
43 views
Write the expressoin in terms of $\log x$ and $\log y \log(\frac{x^3}{10y})$
What is the answer for this? Write the expression in terms of $\log x$ and $\log y$ $$\log\left(\dfrac{x^3}{10y}\right)$$
This is what I got out of the equation so far. the alternate form assuming ...
7
votes
5answers
172 views
What does $x^\pi$ mean? [duplicate]
I was just wondering, what does $x^\pi$ or for that matter, $x$ raised to any irrational number mean? For example, I want to represent $x^2$ then that would mean $x * x$ or if I want to do ...
2
votes
4answers
161 views
For $(x+\sqrt{x^2+3})(y+\sqrt{y^2+3})=3$, compute $x+y$ .
If $(x+\sqrt{x^2+3})(y+\sqrt{y^2+3})=3$, compute $x+y$.
0
votes
1answer
29 views
Calculating how many pieces fit into a given area
Is there any program which calculates how many pieces of an item with different sizes you can put in one area?
For example, I have a sheet of glass with size $3210 \times 2210$mm. Now I have ...
9
votes
5answers
82 views
Reducibility of $x^{2n} + x^{2n-2} + \cdots + x^{2} + 1$
Just for fun I am experimenting with irreducibility of certain polynomials over the integers. Since $x^4+x^2+1=(x^2-x+1)(x^2+x+1)$, I thought perhaps $x^6+x^4+x^2+1$ is also reducible. Indeed:
...
1
vote
2answers
68 views
Pre Calculus Math Equation With Logarithms
Please Help me with this I think i figured out question 1... but I get no solution...
please help me start number 2 or if you can show full solution that be sick thanks.
$\log_{3x}(81)=2$
...
1
vote
0answers
33 views
How to solve this type of equation with posynomial form?
I have an equation with the following form where the goal is to find $x$:
$$ \sum_k c_k x^{\gamma_k} = 1$$
where $c_k, \gamma_k \in \Re^+$ and $\gamma_k > 1$
Alternatively using $y = \log(x)$ I can ...
1
vote
2answers
29 views
Simple algebra simplification question?
Hello everyone I have the following question.
I have the following fraction
$f(x)=-\frac{4}{x^2}+\frac{1}{(x-1)^2}$
But how would I reduce it I know I have to use the multiply the opposite ...
0
votes
5answers
91 views
Finding two numbers when having their sum and product
I have two numbers, their sum is 41 and their product is 238. What are the numbers?
I got during this far in my calculations:
$a+b=41,\quad ab=238,\quad 238=41-b.$
I appreciate answers or tips to ...
1
vote
2answers
36 views
Manually Finding Values of Inverse Trigonometric Functions
I'm trying to solve (for $x$) some problems such as $\arctan(0)=x$, $\arcsin(-\frac{\sqrt{3}}{{2}})=x$, etc.
What is the best way to go about this? So far, I have been trying to solve the problems ...
1
vote
1answer
42 views
A Diophantine equation and decimal digits
Solutions of the Diophantine equation
$a10^n+(a+1) = (2^{m+1}-1)*2^{m+1}$
are
12=3*4,
56=7*8,
67100672=8191*8192.
Are there more solutions/examples like that or a generalization of the ...
2
votes
0answers
41 views
Unique decomposition of $c$ sums of products of $k$ numbers greater than 1, allowing duplicates?
This question differs from Unique decomposition of $c$ sums of products of $k$ prime numbers, allowing duplicates? in that prime number restriction is changed to any number greater than 1.
Suppose ...
0
votes
2answers
37 views
Divide polynomials and conclude:
I have to divide $x^3-a^3$ by $x-a$ and conclude that $x^3 - a^3 = (x-a)*(x^2+ax+a^2)$, but, I'm not sure about how to divide this polynomials, basically doing what I think I should do I will get just ...
1
vote
2answers
37 views
How to solve for a constant
I'm totally stuck at those equations:
$a+b=5$
$ax+b=11$
$a+bx=9$
I need to solve for $x$.
I know this is embarrassing but maybe someone can give me a clue...
1
vote
1answer
60 views
How to find the first three positive values of $\theta=\arctan(3)$?
How would I find the first three positive values of that? All I know is that the equation can also be written as $\theta=\arctan(3)$
2
votes
2answers
67 views
How can I prove this cosine equation?
How to prove that $\cos(90)\cos(\theta)+\sin(90)\sin(\theta)=\sin(\theta)$ ?
0
votes
1answer
49 views
Bicycle Question
In a survey, pupils were asked if they owned a bicycle.
Results:
3/8 of the pupils said ‘Yes’.
5/8 of the pupils said ‘No’.
46 more pupils said ‘No’ than said ‘Yes’.
Altogether, how many pupils ...
1
vote
2answers
56 views
How to find the solutions of $x^3-x^2-x+1=0$
I have to find the solutions of $x^3-x^2-x+1=0$. I know $x = 1$ but how to solve this? I'm lost on the step-by-step resolution.
4
votes
2answers
98 views
Solve $2^{2x} + 9e^{-2x} = 6$ for x using substitution.
This is the equation I have:
$$2^{2x} + 9e^{-2x} = 6$$
I want to solve for x using the substitution method.
I've turned it into
$$4^x+\frac{9}{e^{2x}} - 6 = 0$$
But I do not know what to ...
2
votes
1answer
75 views
Combining differential equations
Can anyone see how to combine the following 3 equations
$$\dot r^2-\dot\theta^2=-\theta^2$$
$$\theta\ddot \theta-2\dot \theta^2=2(\dot r^2-\dot \theta^2)$$
$$\dot r=a \theta^2$$
to get ...
5
votes
4answers
47 views
$\frac{(x + \sqrt{x}) - (x-\sqrt{x})}{\sqrt{x+\sqrt{x}}+\sqrt{x-\sqrt{x}}} = \frac{2}{\sqrt{1+\frac{1}{\sqrt{x}}}+\sqrt{1-\frac{1}{\sqrt{x}}}}$?
According to an example in my text book:
$$\frac{(x + \sqrt{x}) - (x-\sqrt{x})}{\sqrt{x+\sqrt{x}}+\sqrt{x-\sqrt{x}}} = \frac{2}{\sqrt{1+\frac{1}{\sqrt{x}}}+\sqrt{1-\frac{1}{\sqrt{x}}}}$$
I don't see ...
0
votes
3answers
96 views
Proof an Equation is Wrong
“For all numbers $j$ and $k$, $(j + k)^2 = j^2 + k^2$.”
How would you prove this is wrong??
When I times out the bracket with the number it makes $j^2 + k^2$ but that's wrong.
Should I put a ...
2
votes
2answers
102 views
There is enough of gold to…
I read that there is enough of gold layering the surface of planet Earth with 45cm layer of gold.
My question is: Suppose that our planet is a sphere(without mountains and stuff), then how much ...
-4
votes
1answer
111 views
10
votes
2answers
129 views
Basic Mathematics. Trouble with proof, powers and odd numbers.
Greets,
In the exercises, at the end of chapter 1.4, Basic Mathematics, Serge Lang
6) Prove: If $n$ is odd, then $\quad (-1)^n = -1$
How?
The working I did
$$\begin{align}( -1)^n &= ( -1 ...
-7
votes
4answers
67 views
Algebra question. How do you solve? Answer is O.
Calculate value for expression $2(a-2)^2 - 2a(a-3)$ if $a = 4.$ Answer is $0$.
4
votes
5answers
160 views
Checking whether a polynomial of high degree is bijective or not.
Let $P(x)$ be a polynomial of degree $101$. Then $x\mapsto P(x)$ cannot be a one-one onto mapping, i.e., bijective function from $\Bbb{R}$ to $\Bbb{R}$. True or false?
I think is when we take ...
1
vote
2answers
34 views
How to solve this 4 terms equation?
First of all, I tried to search online but I didn't find any explanation.
The following equations, I just don't know how I could solve'em, the question asks for the solutions.
$(x²-4)*(2x-2)*x=0$
...
1
vote
4answers
61 views
Is there an explanation why the reflection of $f(x)$ through y = x is its inverse?
e.g. The function $e^x$ reflected through $y=x$ is $\ln x$. Is this always true OR just in some cases?
1
vote
2answers
63 views
Distance between point and a line - problems simplifying the minimised distance equation
Someone asked how to prove the distance between a point $(x_1,y_1)$ and a line $Ax + By + C = 0$ is:$$\text{Distance} = \frac{\left | Ax_{1} + By_{1} + C\right |}{\sqrt{A^2 + B^2} }$$ The currently ...
2
votes
0answers
42 views
Unique decomposition of $c$ sums of products of $k$ prime numbers, allowing duplicates?
Suppose that there are $n$ different prime numbers. Define procedure a) as following ($k \leq n$ and $k$ fixed): procedure a) for each time, we select one number out of $n$ possible cases and multiply ...
1
vote
2answers
60 views
How to simplify $\frac{(\sec\theta -\tan\theta)^2+1}{\sec\theta \csc\theta -\tan\theta \csc \theta} $
How to simplify the following expression :
$$\frac{(\sec\theta -\tan\theta)^2+1}{\sec\theta \csc\theta -\tan\theta \csc \theta} $$
2
votes
1answer
27 views
Formula for ' constant-power' across 3 sound sources (3-way DJ Crossfader)
First time poster here - thank you profusely in advance for any help you can provide!
Intro/Context
I need help to expand upon an existing mathematical approach to providing 'constant power' in ...
1
vote
1answer
25 views
Satisfying a condition on given quadratic equation
Let $P(x) = x^2 +2bx + c$ be a quadratic form where $b,c$ are real numbers.If $b^2 < c$ , show that $P(x) > 0$ for all $x$ .Is the converse also true?
The value of $x$ after solving the ...
2
votes
1answer
26 views
Show that this type of function is surjective iff it's injective.
Here's a theorem that I think intuitively makes sense, but I was hoping to prove more rigorously:
Theorem: Suppose $|A|=|B|=n$, where $n\in\mathbb{N}$. Consider the function $f:A \to B$. Then $f$ ...
1
vote
2answers
54 views
Determining pendulum rise using trigonometry
Everyone in my math class (including the teacher) is having problems with this trigonometry question:
I am assuming that you halve the pendulum and the bottom of the triangle would be $\frac{1.8}{2} ...
4
votes
1answer
62 views
How to express $\cos(\frac{x}{k})$ and $\sin(\frac{x}{k})$ in terms of $\cos(x)$ and $\sin(x)$, respectively?
How can we express $\cos(\frac{x}{k})$ ($k \in \mathbb{N}$) in terms of $\cos(x)$?
And $\sin(\frac{x}{k})$ in terms of $\sin(x)$?
Edit
Maybe this another question helps. Is there a $T_n(x)$ ...
1
vote
1answer
98 views
Determining the Lipschitz constant
Determine the corresponding Lipschitz constant of $f(t,y(t))=e^{(t-y)/2}$, where $D=\{(t,y) : 0\leq t \leq 1,-\infty<y<+\infty\}$.
2
votes
2answers
58 views
Unable to solve expression for $x$
I'm trying to solve this expression for $x$:
$$\frac{x^n(n(1-x)+1)}{(1-x)^2}=0$$
I'm not sure where to begin (especially getting rid of the $x^n$ part), any hints or tips are appreciated.
0
votes
1answer
61 views
System of nonlinear polynomial equations
How to solve the system
\begin{equation}\frac{24 - x^2}{10x} = -\frac{13 - y^2}{12y}~~~~~~~~~~~~~~(1)\\
x^2 + y^2 + 12 = z^2~~~~~~~~~~~~~~~~~~~~~~(2)\\ \frac{z^2 - x^2 - y^2}{2xy} = -\frac{z^2 - ...
2
votes
4answers
72 views
Basic Mathematics. Trouble with powers and polynomials.
Greets,
I'm hoping to complete the exercises in chapter 1.3 Basic Mathematics, Serge Lang.
The section question is:
Expand the following expressions as sums of powers of $\;x\;$ multiplied by ...
0
votes
1answer
22 views
equation to calculate space on label
I'm a software developer,
I'm working on a rendering PDF report for Android, I have to place a label in the middle of a cell that changes in the X coordinate,
I need to set a cell with some text ...
0
votes
3answers
25 views
Simplify $(p)(\frac{1}{2}(2p-1)^n) + (1-p)(-\frac{1}{2}(2p-1)^n)$ to $\frac{1}{2}(2p-1)^{n+1}$
The expression was simplified in the answer to this question. I'm trying to simplify it but I got stuck. Multiplying all the factors and regrouping didn't work, but maybe I'm doing the wrong ...
0
votes
1answer
48 views
Find function with given properties
Find a smooth function $g: \mathbb{R} \to \mathbb{R}$ that
domain $g$ is $\mathbb{R}$
range of $g$ is a subset of $\mathbb{R^+}$
$g$ is concave.
5
votes
6answers
114 views
Basic Mathematics. Trouble with powers.
Greets,
In Chapter 1.3, Basic Mathematics, Serge Lang, there is the question:
Express each of the following expressions in the form $2^m3^na^rb^s$, where $m, n, r, s$ are positive integers.
b) ...
3
votes
3answers
312 views
Why can I divide a fraction like this?
Suppose I have a fraction: $$\frac{2^n}{2^{2n}+1}$$
I can simplify it to become: $$\frac{1}{2^{n}+\frac{1}{2^n}}$$
Now obviously, this is just dividing both the numerator and the denominator of the ...
