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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22 views

$f(x)$ is a polynomial satisfying $2 + f(x)f(y)=f(x)+f(y)+f(xy)$, find $f(f(2)$), given $f(2)=5.$

If f(x) is a polynomial satisfying $2 + f(x)f(y)=f(x)+f(y)+f(xy)$, find $f(f(2))$, given $f(2)=5.$ ATTEMPT:- $f(f(2))=f(5)$, We can find $f(0)$,$f(1)$ and $f(1/2)$ to be $1,2$ and $5/4$ ...
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0answers
17 views

proportionately

Original amount of money is reduced by \$400 for a tax paid by one of five share holders. At the end the amount is to be divided equally by five share holders, what is the amount for the five shares ...
0
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3answers
45 views

How to solve a system of linear equations using substitution or elimination

I'm supposed to be using the substitution or elimination technique to solve \begin{align*} x + 2y & = 11\\ x - y & = -1 \end{align*} but for some odd reason I continue getting the wrong ...
0
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3answers
372 views

If two polynomials both of n degree have n identical real roots, are they equal? Proof?

CORRECTION: The polynomials don't have to be equal, but one has to be a constant multiple of the other. I ask the question because I saw this fact used in this solution to a problem: Problem: Given ...
2
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4answers
72 views
+100

Precalculus algebra: Greatest term in polynomial

What is the term with the highest power in $$(x^3-2)^{16}-(x^4 + 3)^{12}$$ The textbook answer is $-32x^{45}$, but not sure I understand the method used to handle such a problem. Most answered ...
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0answers
13 views

Out of 100%, if 52% of the population is type X, and 51% type Y…

Out of 100%, if 52% of the population is type X, and 51% type Y but every single type Y person also has a second type X then how many type X people (without type Y) are there? Am I correct to say ...
5
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3answers
430 views

Finding the number of real roots of an unusual(!) equation

How many real roots does the below equation have? \begin{equation*} \frac{x^{2000}}{2001}+2\sqrt{3}x^2-2\sqrt{5}x+\sqrt{3}=0 \end{equation*} A) 0 B) 11 C) 12 D) 1 E) None of these I could not come ...
12
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8answers
4k views

How to become proficient in Calculus?

It has been a while since I wanted to ask this question, but couldn't find a right forum. My question might come across as trivial, but its important to me to find an answer. Let me give you some ...
0
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1answer
18 views

Equation to take out a percentage from a number and have the leftover.

Someone argued with me about the right way to deduct a percentage from a number and have the leftover. Say for example 20% of 2000. Their way: 2000/1.20 = 1,666 (but that seems wrong???) My way: ...
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0answers
10 views

Extrema of quartic functions

I am looking for some general statement how the extrema (or only the maximum) of a quartic function looks like. My quartic polynomial is bounded from below (fourth order coefficient is positive), the ...
12
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3answers
63 views

Prove that $4x^2-8xy+5y^2\geq0$ - is this a valid proof?

I need to prove that $4x^2-8xy+5y^2\geq0$ holds for every real numbers $x, y$. First I start with another inequality, i.e. $4x^2-8xy+4y^2\geq0$, which clearly holds as it can be factorized into ...
0
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1answer
23 views

if john lends $400 for 9months

if john lends \$400 for 9months, peter lends \$300 for 8months, and the interest paid is \$25. How much should john receive? The answer choices are : A.10 B.12 C.15 D.17 E.18 And the correct ...
0
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1answer
375 views

A question using arithmetic progression: How long does it take to fill a rectangular tank?

Water fills a tank at a rate of 150 litres during the 1st hour, 350 litres during the 2nd hour, 550 litres during the third hour and so on. Find the number of hours necessary to fill a ...
5
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1answer
101 views

How can I better solve proofs requiring the introduction of algebraic assumptions?

Today I decided to binge on discrete mathematics after a three year hiatus. I tackled three proofs, and all of them required the introduction of assumptions that seemed to not be found in the givens ...
19
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4answers
284 views

Intriguing Indefinite Integral: $\int ( \frac{x^2-3x+1/3 }{x^3-x+1})^2 \mathrm{d}x$

Evaluate $$\int \left( \frac{x^2-3x+\frac{1}{3}}{x^3-x+1}\right)^2 \mathrm{d}x$$ I tried using partial fractions but the denominator doesn't factor out nicely. I also substituted ...
0
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1answer
26 views

“$\frac{dy}{dx}= -0.11$. It is easy to see that except at one particular point, dy will be of a different size from dx”.What point?

From "Calculus made easy" by Thompson. A ladder of fixed length is against a horizontal wall.Height of ladder is y,distance of base of ladder from wall is x.For positive increment to x, there will be ...
1
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2answers
44 views

How to solve this 3rd degree polynomial?

I looked up the factoring method, but I think this one is calculated using a calculator. With a basic calculator, how do I set this up? $16x^3 - 18x^2 - 2x - 1 = 0$ I factored it to become... ...
7
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1answer
44 views

Sequences of real numbers, arithmetic mean.

Given a sequence of real numbers, a move consists of choosing two terms and replacing each with their arithmetic mean. Show that there exists a sequence of 2015 distinct real numbers such that after ...
4
votes
5answers
209 views

Given that $x^y=y^x$, what could $x$ and $y$ be?

It's not too difficult to figure out that $x$ and $y$ can both be 1, and also $x$ can be 2 and $y$ can be 4 (and vice versa). But I can't rule out if there are other solutions. Does it have anything ...
24
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14answers
906 views

How to explain to a 14-year-old that $\sqrt{(-3)^2}$ isn't $-3$?

I had this problem yesterday. I tried to explain to the kid this: $$\sqrt{(-3)^2} = 3,$$ and he immediately said: "My teacher told us that we can cancel the square with the square root, so it's ...
0
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0answers
65 views

Numbers can be written as $A^B =B^A$ [duplicate]

Of course 16 is the only integer that can be written as $A^B = B^A$ $(4^2 = 2^4)$, where $A$ and $B$ are two distinct integers. My question is: Are there infinitely many (real) numbers that can be ...
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4answers
67 views

Solving $\frac{x+y}{xy}=2$, $\frac{x-y}{xy}=6$

$$\frac{x+y}{xy}=2,\ \ \frac{x-y}{xy}=6$$ I am not understanding how to solve the equation. I tried dividing the whole equation by $xy$, but, that didn't work too. Any hint or help would be much ...
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1answer
17 views

How many different groups of 12 are being in an average of 1.7 groups?

If I have $10$ million people, how do I calculate how many different groups of $12$ with each person being in an average of $1.7$ groups?
0
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0answers
34 views

Find the range of the function : $\frac{1}{\pi}(\sin^{-1}x+\tan^{-1}x) + \frac{x+1}{x^2+2x+5}$ [duplicate]

Problem : Find the range of the function : $\frac{1}{\pi}(\sin^{-1}x+\tan^{-1}x) + \frac{x+1}{x^2+2x+5}$ My approach : Let $g(x) = (\sin^{-1}x+\tan^{-1}x)$ and $h(x)=\frac{x+1}{x^2+2x+5}$ and ...
1
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3answers
32 views

Find the interval on which $ x^{2} - \lfloor x \rfloor - 3 < 0 $ holds.

On what interval does the equation $ x^{2} - \lfloor x \rfloor - 3 < 0 $ hold? My attempt: I tried sketching the graph, but it’s a bit complicated. Is there any other approach?
0
votes
5answers
6k views

Finding an equation of circle which passes through three points

How to find the equation of a circle which passes through these points $(5,10), (-5,0),(9,-6)$ using the formula $(x-q)^2 + (y-p)^2 = r^2$. I know i need to use that formula but have no idea how to ...
0
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2answers
23 views

summations $p$, $q$ For positive real numbers $ a_1, a_2,a_3…, a_{100}$

For positive real numbers $ a_1, a_2,a_3....., a_{100}$, let $$p=\sum_{i=1}^{100} a_i $$ $$q=\sum_{1\le i \lt j \le 100} a_ia_j $$ Then $ q=p^2/2$ $ q^2\ge p^2/2$ $ q\lt p^2/2$ None of these I ...
0
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2answers
26 views

Prove that $-x^2 \leq x^n \leq x^2$ for $-1<x<1, n\in \mathbb N, n \geq3$

Prove that $-x^2 \leq x^n \leq x^2$ for $-1<x<1, n\in \mathbb N, n \geq3$ I have no idea how to do this, I don't even know how to begin. Please help!
2
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0answers
40 views
+50

How to determine when this two variable transformation is invertible?

I am given: $$ X= U \cos(V) \tag{1}\\ $$ $$ Y = U \sin V \tag{2}$$ Now, I need to: a) Give the respective ranges for $U$ and $V$ in order that the transformation defined is one to one. and ...
0
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1answer
54 views

Prove the inequality, $\root3\of4\sin^2(x/2)<3(\sin x+1-x)^{2/3}$

Prove that $$\left(\sin^2{\frac{x}{2}}\right) \cdot \frac{\sqrt[3]{4}}{3} \cdot \frac{1}{{(\sin x + 1 - x})^{\frac{2}{3}}} <1$$
0
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1answer
35 views

How to express a fraction into product of two terms

I want to express the following fraction: $$x^2\left(\frac{y-1}{y+1}\right)^2+x\left(\frac{y-1}{y+1}\right) + 1$$ into $$K(y-c)(y-d),$$ where $c$ and $d$ are zeroes of the above function. How can I ...
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1answer
16 views

Algebraic isolation of integer with power function attached

$$\frac{T_3}{T_2}=\left(\frac{P_3}{P_2}\right)^{\frac{R}{C_p}}$$ I need to find $P_3$, I'm not sure how to deal with the power function? Any help appreciated!
3
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2answers
48 views

Is $\sin^4 x-\cos^4 x = \cos2x$ or is it $-\cos2x=\cos2x$?

A test question I received and got wrong stated that $$\sin^4x-\cos^4x = \cos2x$$ After solving the equation from lower powers of tragicomic functions it came out ...
3
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2answers
40 views

Cyclic Equation. Prove that: $\small\frac { a^2(b-c)^3 + b^2(c-a)^3 + c^2(a-b)^3 }{ (a-b)(b-c)(c-a) } = ab + bc + ca$?

This is how far I got without using polynomial division: \begin{align} \tiny \frac { a^{ 2 }(b-c)^{ 3 }+b^{ 2 }(c-a)^{ 3 }+c^{ 2 }(a-b)^{ 3 } }{ (a-b)(b-c)(c-a) } &\tiny=\frac { { a }^{ 2 }\{ { b ...
0
votes
1answer
34 views

how to find area of segment of only given the radius? [on hold]

I have a thought question it will be tough to explain but here we go. My teacher gave us three circles which each have a radius of 5in. Inside the circle there's a triangle with no angles? Inside the ...
4
votes
1answer
320 views

Why can't I take the square root of both sides when finding the roots of a quadratic equation?

Is $x^2 - x - 12 = 0$ equivalent to $x = \sqrt{x + 12}$? I started with $x^2 - x - 12 = 0$ and made the following changes: $x^2 - x - 12 = 0$ $x^2 = x + 12$ $x = \sqrt{x + 12}$ From here I can ...
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3answers
66 views

not sure how to do this [on hold]

Julie is required to pay a 2 percent tax on all income over 3,000. She also has to pay 2.5 percent on all income over 20,000. She earned more than 20,000 and paid 992.50 what was her total income
2
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4answers
724 views

General questions about Parametric equations

My textbook doesn't explain this very well, so what I want to know is: What is the purpose of parametric equations? What is a parameter? What is the advantage of these equations over a function ...
5
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2answers
91 views

What wolfram does to factor $x^6+x^2+2$?

I am learning polynomials and I am trying to understand what wolfram did to obtain $$(x^2+1)(x^4-x^2+2)$$ from $$x^6+x^2+2$$ It does not show me the step-by-step option in this case and I got ...
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1answer
28 views

Determine the three digits of a number.

A three-digit number in base 7 is expressed by the same numerals, but in reverse order on the basis 9. Determine the three digits.
1
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4answers
494 views

Limit of a Recursive Sequence

I'm having a really hard time finding the limit of a recursive sequence - $$ \begin{align*} &a(1)=2,\\ &a(2)=5,\\ &a(n+2)=\frac12 \cdot \big(a(n)+a(n+1)\big). \end{align*}$$ I proved ...
0
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3answers
45 views

Solving an equation algebraicly

Just wondering, how can ALL the answers be found to the equation: $2^\pi+\pi=2^n + n$ Obviously $\pi$ is a real solution, but how can I get this result; and can I obtain other solutions?
0
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2answers
58 views

Show that $f(a)$ converges after some point

There is a row of 1000 integers. There is a second row below, which is constructed as follows. Under each number $a$ of the first row, there is a positive integer $f(a)$ such that $f (a)$ equals ...
2
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1answer
33 views

5 pears and 1 apple cost as much as 2 pears and 2 apples. If each apple costs $0.75,find the total cost of 100 pears and 450 apples.

I know it's in algebra based on the context and the next question is also algebra. This is what I could do: $y + $0.75 = $x $d + $1.50 = $x I don't know what to do next.
2
votes
2answers
60 views

A problem in definite integral.

What will be the value of $a$ for which the integral $$\int \limits^{\infty }_{0}\frac{dx}{a^{2}+(x-\frac{1}{x})^{2}} =\frac{\pi}{5050}$$ where $a^{2}\geq0$ It seems like a standard integral but ...
1
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2answers
29 views

Help Finding the Equation of a Line with a Given a Point and a Slope

I have been receiving a lot of help today and I really appreciate it, I am nearing the end of my class and need help studying for my final. In my review I came across a problem that i am having ...
0
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0answers
40 views

A nice set of squares. [duplicate]

Are there integers $a, b, c, d$ such that $$a^2+b^2=c^2$$ $$a^2-b^2 = d^2?$$ I have tried by showing that $a^2 = b^2 + d^2$ and thus $a^2+ b^2 = 2b^2+d^2 = c^2$ But how do I show that there are no ...
0
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1answer
30 views

Scaling a grading system

I wasnt quite sure what to call this question. But here is my issue (it might be super simple) I have the following grading system: Some pseudocode ...
1
vote
5answers
147 views

I need help identifying the slope for an equation.

I'm currently having a problem with this $$2x – 5y = 10$$ I'm supposed to write given equation in slope-intercept form and identify slope. I did that, but I thought the answer was $5$, but it keeps ...
1
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4answers
99 views

Can some explain very quickly what $ |5 x + 20| = 5 $ actually means?

I don’t mean to bother the community with something so easy, but for the life of me, I can’t remember how to do these. I even forgot what they were called, and I’m referring to the use of the “$ | $” ...