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
28 views

Do there exist $a,b,c,d,e,f$ such that $ax^2+by^2+cxy+dx+ey+f > 0 \quad\forall 0<x\le 1, 0< y\le 1$ and…

Do there exist $a,b,c,d,e,f$ satisfying: \begin{cases} ax^2+by^2+cxy+dx+ey+f > 0 \quad\forall 0<x\le 1, 0< y\le 1\\ a+b+c+d+e+f \le 1\\ a+d+f \le 0\\ b+e+f \le 0\\ f\le 0 \end{cases}? ...
3
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2answers
64 views

$\frac{1\cdot2^2+2\cdot3^2+\cdots+n(n+1)^2}{1^2\cdot2+2^2\cdot3+\cdots+n^2(n+1)}=\frac{3n+5}{3n+1}$ by Mathematical Induction

Prove by Mathematical Induction: $$\frac{1\cdot2^2+2\cdot3^2+\cdots+n(n+1)^2}{1^2\cdot2+2^2\cdot3+\cdots+n^2(n+1)}=\frac{3n+5}{3n+1}$$ Now by inductive hypothesis: ...
-3
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0answers
16 views

mixing of three type of teas

Three tea, whose prices per kg are respectively 15dollars,25dollars and $30, are to be taken two at a time and mixed in the same proportion so that the resulting mixtures are of equal value. How may ...
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2answers
57 views

If $ \frac{x}{b-c} = \frac{y}{c-a} = \frac{z}{a-b}$, Prove that $x+y+z$ $=0$

Question If $$ \frac{x}{b-c} = \frac{y}{c-a} = \frac{z}{a-b} $$ Prove that $x+y+z$ $=0$ I've attmempted this question by cross multiplying so that $$ x(c-a)(a-b) = y(b-c)(a-b) = ...
-1
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2answers
58 views

What does it mean to say “again” or “finally” in math? [on hold]

According to math rules if we say again, does it mean we are saying repeat previous step? For example: You have $10$ coins. Add $10$ more coins. Add $2$ coins Again $2$. Again $2$. Add $2$. And ...
2
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2answers
43 views

How do I deal with a floor function is a system of equations?

How would one solve an equation with a floor function in it: \begin{cases} y=12(x-\lfloor x \rfloor) \\ x=12(y-\lfloor y \rfloor) \end{cases} Maybe an algebraic method could be used?
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1answer
33 views

Finding a sixth degree polynomial that goes through 8 points

For a summative math research assignment, I will have to find a sixth degree polynomial that would ideally go through the following points: (0, 20.5625) (10, 27.5625) (30, 14.5625) (50, 14.6875) (60, ...
1
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1answer
54 views

shortest distance between the point $(0,-3)$ and the curve $y=1+a_{1}x^2 + a_{2}x^4 + …+a_{n}x^{2n}$

If each $a_{i}>0,$ Then the shortest distance between the point $(0,-3)$ and the curve $$y=1+a_{1}x^2 + a_{2}x^4 + \cdots +a_{n}x^{2n}$$ is $\bf{My\; Try::}$ Let $P(x,y)$ be ant point on the ...
1
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3answers
41 views

Find the smallest positive value taken by $a^3+b^3+c^3-3abc$

Find the smallest positive value taken by $a^3+b^3+c^3-3abc$ for positive integers $a,b,c$. Find all integers $a,b,c$ which give the smallest value. Since it is generally hard to find the minimum ...
2
votes
1answer
41 views

How can I find an output of this function's inverse without graphing?

How can I find $f^{-1}(5)$ where $$f(x)=\frac{27}{\pi}x + \sin x$$ algebraically? Thank you!!
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2answers
70 views

Giving a function for a $1-1$ function

Show that the given intervals have the same cardinality by giving a formula for a $1-1$ function, $f$, mapping the first interval onto the second. $[1,3]$ and $[5,25]$ So I understand that I have ...
0
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0answers
40 views

Finding the symmetries of $f(x)$ given that $f(f(x))=x$

If a function $f$ satisfies the property $f(f(x))=x$, then how would you show that it is an odd function? I tried the following but couldn' t get anywhere. Also, would it have any other symmetries? ...
0
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1answer
33 views

$(X_1+X_2+ X_3 + \cdots + X_n)^2 =$ $?$

$(X_1+X_2+ X_3 + \cdots + X_n)^2 =$ $?$ with $X_i$'s $ \in \mathbb{R}$ Just from computing $(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$ I am guessing the general formula is: $(x_1 + \cdots + ...
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0answers
30 views

How do I form this into an equation? 8\$ per customer, until 50 customers, then profit deceases by .1\$ per customer?

I'm a tutor, tutoring a first year in a business (I'm a physics/engineering student) course. I've had no problems helping her thus far, but I ran across a question in her assignments that I can't ...
1
vote
1answer
19 views

Convert base y to base 10

I have a problem to find base y. The equation given is ($1111011$)gray + ($123$)y + ($211.1$)3 + ($34.4$)$6$ = (CD)$16$ + ($40$)y - ($10010$)BCD. I am able to simplify by converting everything ...
0
votes
2answers
37 views

Is there an easier way to solve big systems of equations?

I have the system \begin{equation*} \begin{cases} 4x^2 - 3xy + 9y^2 = 15,\\ 2x + 3y = 5 \end{cases} \end{equation*} Is there any better way than to substitute $\frac{5-2x}{3}$ in for $y$?
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1answer
9 views
1
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1answer
19 views

cauchy- schwarz inequality b/2a input value

I was watching this video but at 8:05 I don't get why to solve for the function $p(t) = at^2 + bt + c \geq 0$, Sal decides to input $t= \frac{b}{2a}$. Someone made this explanation: $\frac{b}{2a}$ is ...
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2answers
19 views

Matrix Solving Method

Solve my matrix method: $$3y+4x=2xy$$ and $$9y-2x=\frac{5xy}{2}$$ My solution Here: $$3y+4x=2xy$$ Dividing both sides by $xy$ $$\frac{3}{x}+\frac{4}{y}=2$$----(1) Again, $$9y-2x=\frac{5xy}{2}$$ ...
3
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1answer
217 views
+200

The root of summation function

This is a calculation I need for my statistics project Big edit: simplify the function $f(x)$ a lot. Define for $f(x)$, $x\geq 0$, $$ f(x):=\sum_{k=1}^\infty ...
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4answers
129 views

How can efficiently derive $x$ and $y$ from $z$ where $z=2^x+3^y$.

How can efficiently derive $x$ and $y$ from $z$ where $z=2^x+3^y$. Note. $x$,$y$ and $z$ are integer values and $z$ is $4096$ bits integer or even more. For all $z>1$. And if equation be ...
0
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2answers
29 views

Q: Quadratic Division - How to divide two quadratics?

My studies into graphs and models following examples from Khan Academy has helped me on my goal to learn how to chart and model via the quadratic formula However while I have been successful in ...
3
votes
1answer
57 views

Solving for a variable in an inverse function

I was asked to solve this formula for $R_2$: $$\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$$ So I did the following: \begin{align*} \frac{1}{R_2} &= \frac{1}{R} - ...
0
votes
1answer
80 views

Determine the minimum and maximum angles, to the nearest tenth of a degree, that a pipe can make with the horizontal.

For residential drains, a horizontal pipe needs to have a minimum slope of $1/4$ inch per foot and a maximum slope of $1/2$ inch per foot for waste to drain properly. This means that for every ...
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2answers
34 views

how to normalise these values

First of all, i don't know if the correct word is normalise or not, but I'll try to explain my issue. I have a relationship between an object A and an object ...
0
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1answer
51 views

Find the two square roots of a complex without transforming to the trigonometric form. [on hold]

How to solve this? I know it will be solved with De Moivre's Theorem but i don't know how. Find the two square roots of $\frac{-7+3i}{-3-7i}+\frac{-3-7i}{7-3i}$ without transforming to the ...
2
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1answer
47 views

Positive integers $(x,y)$ their sum and product is a perfect square

Is there any genuine approach to find pair of positive integers $(x,y)$ such that both their sum and product is a perfect square? One pair is $(5,20)$ but it looks to me that this question can be ...
1
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2answers
39 views

Can $(\mu_1 - \mu)^2 + (\mu_2 - \mu)^2$ be simplified to remove $\mu?$

$\mu_1$ and $\mu_2$ are means of two groups, while $\mu$ is the overall mean. I feel like this can be done with some basic algebra I've forgotten, but I'm not sure. Here is some more context: $$ f ...
0
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1answer
23 views

how to write as geometric series $\dfrac{A(3s-5)}{(s-3)(3s-5)}+\dfrac{B(s-3)}{(3s-5)(s-3)}$ [on hold]

How would I write $\dfrac{A(3s-5)}{(s-3)(3s-5)}+\dfrac{B(s-3)}{(3s-5)(s-3)}$ as a sum of geometric series?
2
votes
2answers
47 views

Is $f(x) =x^{-1}$ an analytic function?

As a prospective undergraduate who is doing pre-study in preparation for my future endeavours, i recently learnt about analytic functions and would like to know whether $f(x) = x^{-1}$ is analytic in ...
1
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2answers
134 views

How to calculate x in formula $ y =\left( \frac{\frac{17000}{x+400}+8.5}{100}+ 1\right) x $?

I am not even sure how it's officially called (so not sure with tag to give it). As an example if you have a math problem $y = x + 1$. You have a $y$ value, but not $x$. To you revese the problem $y - ...
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votes
1answer
36 views

Is my proof by induction correct?

If $x_1 , x_2,......x_n$ are non-zero elements of a field so is $\prod_{k=1}^n x_k$; and $\left(\prod_{k=1}^n x_k\right)^{-1} = \prod_{k=1}^n x_k^{-1}$. Assume $n = 2$ true; How I did it: First: ...
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2answers
38 views

showing projection is a linear operator

Show that the orthogonal projection is linear. Let $x_i=y_i+z_i$, where $x_i\in X$, $y_i\in Y$, $z_i\in Y^\perp$, and $\alpha,\beta$ be scalars. Then \begin{align}P(\alpha x_1+\beta ...
1
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2answers
23 views

Why coefficients have to be proportional for two quadratic functions to have the same roots?

We have the next two quadratic functions: $ ax^2 + bx + c = 0 $ $ mx^2 + nx + p = 0 $ If $ a/m = b/n = c/p $ then they have the same roots. What is the intuition behind this statement?
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0answers
30 views

Can a sum of trigonometric functions equal a constant for all inputs?

Let $r_1,...,r_n$ and $\phi_1,...\phi_n$ be real numbers. Consider the following sum: $S=\sum\limits_{k=1}^{n}r_k\sin(\phi_k+k\alpha)$ Suppose $S$ is constant for all $\alpha \in R$. Does it ...
0
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2answers
30 views

Expanding logarithm of function

Is there a way (there has to be), I can expand an expression like this? $$\log_2 (3f(n)^n)$$ P.S. This part of an assignment I'm working on, please do not give solutions
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0answers
47 views

Find the positive integers $\overline {abc}$ such that $\frac{1}{a} +\frac{1}{b}+\frac{1}{c}$=$\frac{\overline {1b}}{\overline {1a}}$ [on hold]

Find the positive integers $\overline {abc}$ such that $$\frac{1}{a} +\frac{1}{b}+\frac{1}{c}=\frac{\overline {1b}}{\overline {1a}}.$$ Can you help me with a solution without to consider the case ...
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0answers
42 views

Find $a,b,c \in \{1,2,..,9\}$ such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{10+a}{10+b}$ [on hold]

Find $a,b,c \in \{1,2,..,9\}$ such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{10+a}{10+b}.$$ It seems to be easy but I want a smart solution.
1
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1answer
25 views

How to expand $x_1^3 + x_2^3$ with the parameters of quadratic equation

Given: $X_1$ and $X_2$ are the roots of the equation $ax^2+bx+c = 0$ $a\neq 0$ expand $X_1^3 + X_2^3$ using the parameters a,b and c Here's what I tried to do: $X_1^3 + X_2^3 = $ $(X_1\cdot ...
0
votes
1answer
34 views

how to calculate the similarity between two items in this case

I have an item A (Symphony Impromptu No. 1 for Frederic Chopin) and i want to know if it is more similar to another item B ...
7
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3answers
156 views

How do you find the maximum value of $|z^2 - 2iz+1|$ given that $|z|=3$, using triangle inequality?

Problem: How do you find the maximum value of $|z^2 - 2iz+1|$ given that $|z|=3$, using triangle inequality? My attempt: $$|z^2 - 2iz+1|\le|z|^2+2|i||z|+1$$ $$\implies |z^2 - 2iz+1|\le16$$ ...
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0answers
23 views

The volume of a specific rectangular prism is represented by $V(x) = -2x^3 + 10x^2 + 300x$. How do roots, vertices, and end behavior apply?

The volume of a specific rectangular prism is represented by $V(x) = -2x^3 + 10x^2 + 300x$, where $x$ is the height of the prism. How do roots, vertices, and end behavior apply? How is the graph ...
0
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0answers
38 views

On summation of series [on hold]

Consider the equality of summations $\sum_{a} f(a) = \sum_{a}f(1-a)$ where both sums are convergent. What conditions need to be satisfied such that $f(a) = f(1-a)$ for all $a$, where $a$ is a ...
0
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2answers
29 views

What is one possible distance (in km) at which I live from Arun’s place?

Michael lives $10$ km away from where I live. Ahmed lives $5$ km away and Susan lives $7$ km away from where I live. Arun is farther away than Ahmed but closer than Susan from where I live. From the ...
3
votes
2answers
56 views

Finding the minimum of $x^2+y^2$ when $(x^2y-xy^2)(x^3-y^3)=x^3+y^3$

If $x,y \in \mathbb {R}$, find the minimum of $x^2+y^2$ when $(x^2y-xy^2)(x^3-y^3)=x^3+y^3$ and $xy>0$. This problem was inspired by a problem which asked if $x,y \in \mathbb {R}$ and $xy \neq ...
1
vote
1answer
47 views

Find “almost inverse” of positive definite bilinear form

Let $A$ be a positive definite $d \times d$ matrix, and define $A(x,x)=x^TAx$. Let $x$ be a point such that $\vert x^T\xi\vert^2\leq \xi^T A\xi$ for all $\xi\in\mathbb{R}^d$. Is this somehow ...
1
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2answers
37 views

Find all $x$ such that $8^x(3x+1)=4$

Find all $x$ such that $8^x(3x+1)=4$,and prove that you have found all values of $x$ that satisfy this equation. My effort Rewriting the equation I have \begin{array} 22^{3x}(3x+1)&=2^2 \\ ...
0
votes
0answers
23 views

Irrational roots conjugate theorem

This theorem seems pretty clear cut at first, but i have read a lot of queries about it. I have found out that if a cubic has only $1$ irrational root, then it cannot be expressed in the form $a + ...
3
votes
1answer
22 views

Choosing a combination of books, under given restrictions.

Mary has on her bookshelf 5 novels, 5 biographies, and 8 textbooks. Mary decides to take three novels and four non-fiction books with at least one of the non-fiction books a biography. How many ...
4
votes
3answers
430 views

Find root of the equation

Find maximum root of the equation $$x - \frac{1000}{\log 2} \log x = 0$$ It locates between $13746$ and $13747$, but I want to find right solution not using graphing calculators. Thanks in advance.