# Tagged Questions

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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### 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 $x \, \mathrm{cm}$. ...
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### 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
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### 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?
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### 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?
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### 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 ...
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### Trouble simplifying the following expression. [closed]

Let $x = t \cos(2t)$ and let $y = t \sin(2t)$. Now show the following equation is true. $$-200xe^{-x^2-y^2} (\cos2t - 2t \sin2t) - 200ye^{-x^2-y^2} (\sin2t +2t \cos2t) = -200te^{-t^2}$$ ...
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### $2k-1$ 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: "...
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### 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 ...
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### 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 $4$: $f(x)=\frac{x^2 + mx + 1 }{x^2-x+1}$.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
### 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....
### 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 $2$ ...