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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-1
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2answers
52 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 ...
0
votes
2answers
49 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) = ...
0
votes
1answer
32 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, ...
2
votes
2answers
41 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?
1
vote
1answer
23 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 0\\ a+d+f \le 0\\ b+e+f \le 0\\ f\le 0 \end{cases}? ...
1
vote
3answers
39 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 ...
0
votes
0answers
39 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
votes
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 + ...
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!!
0
votes
2answers
36 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 $y$?
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 ...
1
vote
2answers
48 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: ...
0
votes
1answer
9 views

Understanding the steps taken in a calculation of the maximum profile likelihood of a simple ODE, given some data

I'm trying to understand a calculation made in a paper (section 2 from the supplementary contents of ...
0
votes
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, $$9y-2x=\frac{5xy}{2}$$ ...
1
vote
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 ...
0
votes
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
votes
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 ...
0
votes
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
vote
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
votes
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 ...
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
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: ...
1
vote
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?
1
vote
0answers
29 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
vote
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
vote
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
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
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 ...
-1
votes
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.
-2
votes
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
votes
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
votes
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 ...
0
votes
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 ...
1
vote
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
22 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
votes
1answer
55 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
38 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
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 ...
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 ...
0
votes
2answers
42 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
13 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
votes
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
64 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
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.
1
vote
2answers
26 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
41 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
56 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
31 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
votes
1answer
40 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.