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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1answer
30 views

Area in terms of $x$.

A wire of $80 \, \mathrm{cm}$ is arranged to form $3$ sides put against a wall forming a rectangle. The longest sides of the rectangle is the wall and a piece of wire with length $x \, \mathrm{cm}$. ...
1
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2answers
21 views

I need help for steps in how to solve for $L$ [closed]

$$-(1-L)^{-\frac{1}{2}}L^{\frac{1}{2}} + (1-L)^{\frac{1}{2}}L^{-\frac{1}{2}}=0$$ Thanks in advance, I've been stuck on this for a while. Chris
4
votes
1answer
77 views

Is there an 'interesting' way to derive this expression?

So I was asked to prove the following term is equal to $2016$: $$ \left( \frac{251}{ \frac{1}{ \sqrt [3] {252} - 5 \sqrt [3] {2} } -10 \sqrt [3] {63} } + \frac {1} { \frac {251} { \sqrt [3] {252} +5 ...
1
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1answer
30 views

How would tetration work for non integer numbers.

Can you even do these and how would you do them? How does tetrations algebraically work? $$^{.5}x=?$$ $$^{-1}x=?$$ $$^ix=?$$ Is there such a number like e that converges? $$^xd=(some/equation/with/x)$$...
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0answers
11 views

Determine the operation based on the conditions given below

\begin{align} f(c, d)&= a;\\ g(c, d)&= b;\\ h(a, b, c)&= d. \end{align} The functions $f$, $g$, $h$ are defined for all $a,b,c,d\in\mathbb R$. For instance: $h$ can be Division; $a$, $b$, ...
1
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4answers
73 views

How to simplify this expression

I'm trying to find a way to simplify $\sqrt{5+\sqrt{24}}$. I know that this expression is equivalent to $\sqrt{2}+\sqrt{3}$ because they are both roots of the equation: $x^4-10x^2+1$ (and the decimal ...
0
votes
1answer
20 views

Standard form of an Ellipse Given…

So I'm currently stuck on how to get the standard form of the equation of the ellipse given the characteristics Vertical Major Axis and passes through the points ( 0,6 ) and ( 3,0 ) Any Ideas? ...
0
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1answer
66 views

Discriminant of Cubics and Math Olympiad

Let $a,b,c$ be distinct nonzero real numbers. If the equations $E_1: ax^3+bx+c=0, E_2: bx^3+cx+a=0$ and $E_3: cx^3+ax+b=0$ have a common root, prove that at least one of these equations has three real ...
4
votes
3answers
147 views

Solve 3 exponential equations $z^x=x$, $z^y=y$, $y^y=x$ to get $x$, $y$, $z$.

The main question is : $z^x=x$, $z^y=y$, $y^y=x$ Find $z$, $y$, $x$. My method : I first attempted to get two equation for the unknowns $x$ and $y$. We can happily write : $z=x^{1/x}$ and $z=y^{...
0
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0answers
51 views

Simplify $F(x) = \exp[-\ln^2x^h]$

I was wondering if the expression $F(x) = \exp[-\ln^2x^h]$ can be simplified even further? As you can see, the $\ln$ which is the natural logarithmic function is raised (and not its argument) to power ...
-4
votes
0answers
31 views

Solve for the conditions given below [closed]

\begin{align} f(c, d)&= a;\\ g(c, d)&= b;\\ h(a, b, c)&= d. \end{align} The functions $f$, $g$, $h$ are defined for all $a,b,c,d\in\mathbb R$. For instance: $h$ can be Division; $a$, $b$, ...
2
votes
1answer
83 views

What is the value $f(-4)$ in the under function such that $f(x)+f(\frac1x)=\frac{x^2-12x+1}{2x}.$

Let $f$ is a function such that $$f(x)+f(\frac{1}{x})=\dfrac{x^2-12x+1}{2x}.$$ Then what is the value $f(-4)=$?
0
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2answers
38 views

Does this set of coordinates result in a curve?

Coordinates: (0,0), (3,3), (6,4.5), (9, 5.25) If this is a curve is there a formula for determining the y value for any given x within the range 0 to 9?
2
votes
2answers
41 views

Domain of $f(x)=x^{\frac{1}{\log x}}$

What is the domain of $$f(x)=x^{\frac{1}{\log x}}$$ Since there is logarithm , the domain is $(0 \: \infty)$ But the book answer is $(0 \: \infty)-\{1\}$ but if $x=1$ $$f(x)=1^\infty=1$$ So is it ...
1
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2answers
24 views

Why does this equality work when k, N, and r are all positive?

The expression is $\frac{r^N - \left( r-\epsilon \right )^N}{r^N}=1 - \left ( 1- \frac{\epsilon}{r} \right )^N$. I understand where the first $1$ comes from, but where does the $\left ( 1- \frac{\...
4
votes
2answers
202 views

How to approach general solutions to functional equations of multiple variables

I understand the concept of a function, broadly speaking, but when it comes down to solving general functional equations, I sometimes find it difficult to wrap my head around the problem at hand. For ...
5
votes
1answer
52 views

Bound on $c-b$ for $a^n+b^n=c^n$

Let $a\leq b\leq c$ be positive real numbers and $n$ positive integer with $a^n+b^n=c^n$. Prove that $c-b\leq(\sqrt[n]{2}-1)a$. The desired inequality can be written as $c-b+a\leq \sqrt[n]{2}a$. ...
1
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2answers
39 views

Is it correct this reasoning?

Let $E,F$ be reals vector space. Since (1) $\dim (E\times F)=\dim E + \dim F$ (2) $\dim\ \text{Hom}(E,F)=\dim E\cdot \dim F$ Given $r>0$ integer, is it true that: $$\text{Hom}(E\times \stackrel{(...
1
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5answers
94 views

How to decompose $x^3-1$

I need to decompose $x^3-1$, I know the Binomial theorem, and finding roots of a polynomial, how should I approach this?
1
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1answer
65 views

What is the equation of this graph?

This will sound very dumb, but I want $1000$ coordinates of this shape: How can I do that?
3
votes
2answers
65 views

Represent $\dfrac{\lambda_1^M-\lambda_2^M}{\lambda_1-\lambda_2}$ in terms of $\lambda_1+\lambda_2$ and $\lambda_1\lambda_2$

I have a problem as follows: Let $\lambda_1, \lambda_2$ are roots of the equation $\lambda^2-a\lambda+b=0.$ It can be proved easily (by induction for example) that the quantity $$\dfrac{\lambda_1^M-\...
0
votes
1answer
15 views

Algebra and summation question

${(1+\frac{q_jr^i}{1-q_j})}^{-1}=\sum_{k=1}^{\infty} (-1)^{k-1}(\frac{q_jr^i}{1-q_j})^{k-1}$ What rule is being used to go from the LHS to the RHS? My knowledge in maths is first year undergrad, but ...
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votes
0answers
23 views

Wanting to reverse an equation to determine d

I am wanting to please get some assistance to reverse this calculation so that I can determine d based on a varying Q. Apologies for the shockingly written and presented equation.. $$Q = \frac {4....
0
votes
1answer
49 views

When are we permitted to multiply or divide both sides of an equation by a constant?

For example, let's consider the quadratic equation $-3x^2 + 6x -2 = 0$. Multiplying both sides by $-1$, we get the equation $3x^2 - 6x +2 = 0$. The graph of the above equations are different even ...
-3
votes
1answer
46 views

Trouble simplifying the following expression. [closed]

Let $x = t \cos(2t)$ and let $y = t \sin(2t)$. Now show the following equation is true. $$-200xe^{-x^2-y^2} (\cos2t - 2t \sin2t) - 200ye^{-x^2-y^2} (\sin2t +2t \cos2t) = -200te^{-t^2}$$ ...
0
votes
1answer
25 views

$2k-1$ is an odd integer if $k$ is an integer

I am working on this advanced power rule problem: This is the image of the problem I understand everything up until step 4 in the problem hint. I am getting stuck with the statement that says: "...
5
votes
0answers
72 views

How to find area of a polygon built on the roots of a given polynomial?

How to find the area of a (maximum area convex) polygon, built on the roots of a given polynomial in the complex plane? For example, consider the equation: $$2x^5+3x^3-x+1=0$$ It has one real and ...
0
votes
1answer
29 views

Find parameter m knowing that the values of the function are in a interval of length 4

Please give me a hint on how to find the parameter $m$ knowing that the function values are in an interval of length $4$: $f(x)=\frac{x^2 + mx + 1 }{x^2-x+1}$.
0
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5answers
48 views

Inverse Equation of the Given Equation

Having a bit of a problem getting the inverse of the following equation: $$f(x) = \sqrt{9-x^2}$$ I had an answer which was equal to $3-x$ but when I used sites like Mathway and Wolfram to check my ...
1
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1answer
77 views

Find the values of $b$ for which the equation $2\log_{\frac{1}{25}}(bx+28)=-\log_5(12-4x-x^2)$ has only one solution

Find the values of 'b' for which the equation $$2\log_{\frac{1}{25}}(bx+28)=-\log_5(12-4x-x^2)$$ has only one solution. =$$-2/2\log_{5}(bx+28)=-\log_5(12-4x-x^2)$$ My try: After removing the ...
1
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2answers
53 views

Find the value of $P(1)$

Let $P (x) = x^2 + bx + c$, where $b$ and $c$ are integer. If $P(x)$ is a factor of both $x^4 + 6x^2 + 25$ and $3x^4 + 4x^2 + 28x + 5$, find the value of $P(1)$. I am not being able to solve ...
1
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1answer
34 views

Quadratic Equation Based Problem:Prove either $a = 2l$ & $b = m$ or $b + m = al$

If by eleminating $x$ between the equation $x² + ax + b = 0$ & $xy + l (x + y) + m = 0$, a quadratic in $y$ is formed whose roots are the same as those of the original quadratic in $x$. Then ...
0
votes
1answer
28 views

What's the relation between earth coordinates and angles?

I've been looking for an answer for a specific question, a part of my question maybe related to this: Calculate the angle of a vector in compass (360) direction However, my question is more specific, ...
1
vote
0answers
43 views

Proving $a^ma^n=a^{m+n}$ by induction when $n$ or $m$ is negative (or both)

Suppose we have already proved this exponent law for when $m,n\in\mathbb{Z^+}$ as in here. Also suppose $x^{-n}=\frac{1}{x^n}$ is given as a definition. Let $m=-\lambda$ and $n=-\gamma$, where $\...
0
votes
2answers
51 views

Finding the sum of $\cos45°$ + $i\cos135°$ + … + $i^{n}\cos(45+90n)°$ + … + $i^{40}\cos3645°$

My question is as follows: If $i^{2}$ = -1, find the value of $$\cos45° + i\cos135° + \ ...\ + i^{n}\cos(45+90n)° + \ ...\ + i^{40}\cos3645°$$ without the aid of a calculator. In terms of my attempts ...
1
vote
2answers
48 views

Prove that this is one-one, but not onto $\Bbb R$.

$\Bbb R$ stands for real numbers. $ f(x) = \begin{cases} 2-x, & \text{if $x \le 1 \qquad \text{is one to one but not onto } \Bbb R $ } \\ \frac{1}{x} , & \text{if $x >1$ } \end{cases}...
2
votes
3answers
128 views

Evaluate $\cos 36^\circ - \cos 72^\circ$ without the aid of a calculator [duplicate]

I have a quick question about a difficult trigonometric functions problem that I have been assigned. The problem is as follows: Evaluate $$\cos36° - \cos72°$$ without the aid of a calculator. In terms ...
2
votes
1answer
66 views

Inequality on a sequence of $n$ reals whose sum is $0$

Consider $n\geq3$ real numbers $a_1,a_2,\dots ,a_n$ satisfying $a_1+a_2+\cdots+a_n=0$ and $$2a_k \leq a_{k-1}+a_{k+1}$$ for all $2\leq k\leq n-1$. Prove that $$|a_k|\leq\frac{n+1}{n-1}\,\max\big\{|a_{...
0
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0answers
30 views

calculating moments in a table

I am trying to calculate the moments in a data list position data 1 15 2 22 3 5 4 2 5 1 to find out where in the list is ...
0
votes
2answers
57 views

Why can't z = 0 in this rational expression?

I came across this expression, which I was asked to simplify and then choose the number that would make the expression undefined: $$\frac{17z^3+17z^2}{34z^3-51z^2}$$ I simplified the expression to $\...
3
votes
8answers
90 views

How to prove the inequalities between $20^{70^2},30^{60^2},40^{50^2}$

Let $$M=\{ 20^{70^2}, 30^{60^2},40^{50^2}\}$$. What number is the greatest and which is the smallest? I thought about beginning by assuming certain inequalities and trying to prove them, for example: ...
0
votes
1answer
34 views

Sides of triangle are in A.P., find its perimeter

The sides of a triangle are in Arithmetic Progression $(A.P.).$ If the smallest angle of the triangle is $\alpha$ and largest angle of the triangle exceeds smallest angle by $\beta$ , then what is the ...
2
votes
2answers
50 views

Find a point on $y=\frac{1}{x^2}$ such that $y'=16$

I'm very new in this forum and I hope I don't ask something silly, which is asked many times before. I have to answer this question: Find the coordinates of the point(s) at which the curve has ...
2
votes
2answers
27 views

How to Solve $2.3856 + \log r = \log(364r - 363)$

I am solving geometric sequence and series problem, but got stuck on the logarithm part. we haven't tackled logarithm yet so this is supposed to be a challenge problem. can anyone advise on how to ...
1
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3answers
45 views

Progressions modulo $n$

I don't understand how to do these 2 tasks: 1) Prove that any arithmetic progression modulo $n$ has a period that divides $n$. 2) Prove that any geometric progression modulo a prime number $p$ has a ...
0
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0answers
25 views

Percentage of votes received in an election

In an election, $70\%$ of males were registered voters and $40\%$ females were registered voters, all registered males casted their votes. But only $65\%$ registered females casted their votes. If ...
1
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2answers
33 views

How to solve this inequality problem?

Given that $a^2 + b^2 = 1$, $c^2 + d^2 = 1$, $p^2 + q^2 = 1$, where $a$, $b$, $c$, $d$, $p$, $q$ are all real numbers, prove that $ab + cd + pq\le \frac{3}{2}$.
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votes
1answer
69 views

Is there a basis which spans the real numbers?

Is there a finite set of real numbers $S=\{a_1, a_2, ..., a_n \}$ such that every real number can be written as a linear combination (with integer coefficients) of the elements of $S$? If no, is there ...
2
votes
1answer
58 views

Sum of series $1−ω^2+ω^4−ω^6+ω^8−ω^{10}+ω^{12}+⋯+ω^{600}−ω^{602}+ω^{604}$

I need to find sum of the series involving cube roots of unity $1−ω^2+ω^4−ω^6+ω^8−ω^{10}+ω^{12}+⋯+ω^{600}−ω^{602}+ω^{604}$. Found it in an old test paper. I applied Geometric Progression Sum Formula....
1
vote
1answer
32 views

The greatest common divisor of $(O_n, T_n+2)$ where $O_n$ and $T_n$ are the oblong and triangular numbers respectively.

Suppose that $T_n$ is odd. Can we find infinitely many $n$ such that $(O_n, T_n+2)=1$? Is it trivial and obvious? My hunch based on some hand calculations is to look at $n$ congruent to $0$ or $2$ ...