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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0
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2answers
10 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, $$9y-2x=\frac{5xy}{2}$$ ...
0
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1answer
12 views

cauchy- schwarz inequality b/2a input value

I was watching this video https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/dot_cross_products/v/proof-of-the-cauchy-schwarz-inequality but at 8:05 I don't get why to solve for the ...
0
votes
1answer
48 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
votes
1answer
45 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 ...
0
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1answer
22 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?
1
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2answers
38 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
votes
2answers
26 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 ...
2
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2answers
42 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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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: ...
1
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2answers
22 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
27 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 ...
1
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2answers
37 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
vote
1answer
24 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
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
0
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1answer
32 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 ...
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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.
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0answers
22 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 ...
-1
votes
0answers
37 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
votes
2answers
28 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 ...
0
votes
2answers
33 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 ...
1
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2answers
35 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 \\ ...
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0answers
21 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 + ...
0
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1answer
52 views

Solve this system of equations without calculator

$$2a +4b +3c +5d +6e=37$$ $$4a +8b +7c +5d +2e=74$$ $$-2a -4b +3c +4d -5e=20$$ $$a +2b +2c -d +2e=26$$ $$5a -10b +4c +6d +4e=24$$ find $a,b,c,d,e$ I tried solving the system of equations above but ...
0
votes
2answers
37 views

What happens to the graph of $f$?

I'm trying to figure out what happens to the graph of $f$ in the following to situations: $f = f(|x|)$ $f = f(\frac{1}{x})$ For the first, I know if $f = |f(x)|$ then the points below the $x$ ...
3
votes
2answers
55 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 ...
3
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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 ...
0
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2answers
41 views

Why is $\cos\left(\frac{3\pi}{2}-t+2k\pi\right) = -\sin(t)$ [on hold]

Why is this true? $$\cos\left(\frac{3\pi}{2}-t+2k\pi\right) = -\sin(t)$$
-3
votes
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 ...
0
votes
1answer
12 views

Write the particular equation expressing cost in terms of miles traveled

To take a taxi in downtown St. Louis, it will cost you $3.00$ to go a mile. After $6$ miles, it will cost $5.25$. The cost varies linearly with the distance traveled.
2
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0answers
39 views

Expanding trigonometric functions with binomial expansion

I was challenged to take $\cos^{\pi}(\pi)$ and expand it using binomial expansion and $\cos(x)=\frac{e^{xi}+e^{-xi}}2$, which I tried: $$\cos^{\pi}(\pi)=\left(\frac{e^{\pi i}+e^{-\pi ...
1
vote
3answers
63 views

Where is the mistake in solving the inequality?

Where am I going wrong in solving this inequality? $$\frac{p-\sqrt{9p-20}}{p-5}<2$$ On cross multiplying and squaring to remove the square root,I get the inequation $p^2-29p+120<0$ Which ...
4
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3answers
426 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.
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2answers
25 views

Condition for roots to lie in certain intervals

The set of values of $p$ such that both the roots of the equation $$f(x)=(p−5)x^2−2px+(p−4)=0$$ are positive and one of the roots is less than $2$ and the other root lies between $2$ & $3$ ...
0
votes
5answers
40 views

Equation $\log(x^2+2ax)=\log(4x-4a-13)$ has only one solution; then exhaustive set of values of $a$ is

Equation: $$\log(x^2+2ax)=\log(4x-4a-13)$$ It has only one solution; then exhaustive set of values of $a$ is ?? I don't even know where to begin The answer is : $$(-13/4,-13/12) \cup [-1]$$
1
vote
1answer
54 views

Prove ${20n \choose 10n}\ge {2n-1 \choose n-1}^{10}$

As the title says, I can't prove that, no matter what I try. What I've tried so far: induction: seemed the most obvious method, since we already had a lot of tasks with it, but using the esimates ...
1
vote
3answers
30 views

For how many days will the food last in garrison?

A garrison has sufficient food for $75$ soldiers for a period of $90$ days. After $10$ days, one third of the soldiers leave. After another $10$ days, $5$ soldiers return, From this day on, ...
0
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1answer
39 views

Divisibility test for 720 [on hold]

Use the divisibility test where possible to list all factors of 720 Please show further examples where appropriate, thank you.
0
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1answer
24 views

Solving for numerator in equation with logarithms (Activation Energy Equation)

I'm having trouble solving for k1 in this equation: ln(0.286/k1) = (100000/8.314)(1/500 - 1/490) The right side should equal 0.491, which I can calculate just fine, but then the left side gives me ...
2
votes
1answer
34 views

Complex derivative numerically using real $h$ and imaginary $h i$?

I want to find numerically (the functional expression might become too complicated) the derivative of a complex function (to use it in a Newton-algorithm). Can I simply do something like $$ \frac ...
0
votes
1answer
32 views

$x^J = y$, $J = 2.455\ldots$ What's the rest of $J$?

I have a problem where I need to know what J is. I do x^J and get y. For example, if I do 5^J, I would want to get 55 as y. Same with 4^J = 30. When J is 2.455, it works up to 4 only! I need for ...
2
votes
2answers
46 views

Method for solving the equation $3 + 5x^{1/2} = 2x$?

As I'm not even sure what type of problem this is, so I can't research it. $$3+5x^{1/2}=2x$$ My Question: Could I get an explanation (with example) of how to solve the above equation? Or direct me ...
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votes
1answer
29 views

Notational problem [on hold]

Please, how do I write the following as a combination of a sum and a product: $$ (c-a_1)(c-a_2)(c-a_3)b_1 + (c-a_2)(c-a_3)b_2+(c-a_3)b_3 ?$$ Also, how can I generalize it?
2
votes
1answer
35 views

Are these two events $A$ and $B$ independent?

Abe and Bernard are dealt five cards each from the same $52$ card deck. Let $A$ be the event that Abe gets a flush (five cards of the same suit) and $B$ be the event that Bernard’s five cards are of ...
4
votes
1answer
27 views

$p(X)$, $P(Y)$, $p(Z) > 0$ and every pair of these events is independent, then $p(X \wedge Y \wedge Z) > 0$?

Is the following statement true or not? Let $X$, $Y$, $Z$ be $3$ events in the same sample space such that $p(X)$, $P(Y)$, $p(Z) > 0$ and every pair of these events is independent. Then $p(X ...
1
vote
1answer
78 views

Why does $\sum\limits_{n=0}^{+\infty} z^n=\frac{1}{1-z}?$

Having $f(z)=\sum\limits_{n=0}^{+\infty} \frac{1}{n!}z^n$ I had to find what $\sum\limits_{n=0}^{+\infty} \frac{1}{n!}z^n\sum\limits_{n=0}^{+\infty} \frac{D_n}{n!}z^n=\sum\limits_{n=0}^{+\infty} ...
1
vote
2answers
12 views

Find parallel line value

I've an homework problem that i'm unable to find the right answer. The problem is: The line $tx + sy = 2$ goes through point $(2,1)$ and is parallel to line $y = 8 -3x$, find the value of $t^2 + ...
0
votes
1answer
25 views

How do you calculate the change in thickness of a cylinder, if you shave off a flat section?

I have a piece of steel, cylindrical (hollow), 200mm outside diameter with 160mm inside diameter (...
7
votes
3answers
155 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: $$|z^2 - 2iz+1|\le|z|^2+2|i||z|+1$$ $$\implies |z^2 - 2iz+1|\le16$$ ...
0
votes
1answer
26 views

What order do I write algebraic math problems?

My math teacher is a little bit picky on the order we write our simplified algebraic math problems, and I forgot to take notes on this. Let's say you have the problem $2(y + 3) - 4x$. Would I write ...
0
votes
2answers
29 views

How to solve a quadratic inequality that acts like a quadratic equality?

This will be largely a trivial question. But how do I solve an inequality like this: $3x^4 - 4x^2 + 1>0$ ? Of course, I can treat it like a quadratic inequality by saying $t=x^2$ So I can solve ...