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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1answer
25 views

How to find radius of hemisphere in applied problem

If the stem of a mushroom is modeled as a right circular cylinder with diameter $1$, height $2$, its cap modeled as a hemisphere of radius $a$ the mushroom has axial symmetry, is of uniform ...
0
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3answers
30 views

Proof of divisibility: $17 \mid 3 \cdot 5^{2015} + 2^{2017} \cdot 5^{670}$ [on hold]

As the title says, prove that $3 \cdot 5^{2015} + 2^{2017} \cdot 5^{670}$ is divisible by $17$.
2
votes
1answer
21 views

Calculate $\sqrt{x^2+y^2+2x-4y+5} + \sqrt{x^2+y^2-6x+8y+25}$, if $3x+2y-1=0$

As the title says, given $x,y \in \mathbb{R}$ where $3x+2y-1=0$ and $x \in [-1, 3]$, calculate $A = \sqrt{x^2+y^2+2x-4y+5} + \sqrt{x^2+y^2-6x+8y+25}$. I tried using the given condition to reduce the ...
1
vote
1answer
14 views

Divisibility: $60 \mid (2x-y)(2y-z)(3z+2x)$, if $8x-10y+27z=0$

As the title says, given $x,y,z \in \mathbb{Z}$, where $8x-10y+27z=0$, prove that $(2x-y)(2y-z)(3z+2x)$ is divisible by $60$. I tried to bring the formula in a format of $(\cdots)(8x-10y+27z) + ...
2
votes
1answer
41 views

Find all functions $F(x)$ for which $F (x) + F ((x − 1)/x) = 1 + x$

Let $F (x)$ be the real-valued function defined for all real $x$ except for $x = 0$ and $x = 1$ and satisfying the functional equation $F (x) + F ((x − 1)/x) = 1 + x$. Find $F (x)$. This ...
3
votes
3answers
48 views

Basic algebra problem: $ \frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x^2}-\frac{1}{y^2}} $

Basic algebra problem I can't seem to figure out: $$ \frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x^2}-\frac{1}{y^2}} $$ $x,y \in \mathbb{R}, x^2 \neq y^2, xy\neq0$. Now I know the result is: ...
1
vote
2answers
34 views

Probably very basic Euclidean geometry; Why is the following expression valid for a point along a straight line?

I am looking at constructible points in abstract algebra, particularly in $\mathbb{C}$. Alongside a proof of a theorem, I came across this expression which I cannot work out how it's been derived. It ...
1
vote
3answers
40 views

How to simplify this equation to solve for m?

It has been way too many years since high school. How can I simplify this equation to solve for m: $\frac{x}{c+pm}=m$ I got to $x = cm + pm^2$ and I don't know how to get any further. I wish this ...
0
votes
0answers
30 views

Squaring the square root of $x$ vs the square root of $x^2$ [duplicate]

While reviewing this algebra concept yesterday I realized I could not justify why this $\sqrt{x^2}=|x|$ is different from this $(\sqrt{x})^2=x$. My confusion stems from the fact that radicals can be ...
-3
votes
5answers
78 views

Catherine is now twice as old as Jason but 6 years ago she was 5 times as old as he was. How old is Catherine now? [on hold]

This is an IQ question. "Catherine is now twice as old as Jason but 6 years ago she was 5 times as old as he was. How old is Catherine now?" How to solve such questions? I think their combined age ...
2
votes
3answers
37 views

Factorise Algebraic Expression

Background: I came across the following problem in class and my teacher was unable to help. The problem was factorise $x^6 - 1$, if you used the difference of 2 squares then used the sum and ...
-1
votes
0answers
15 views

Graph transformations (g in terms of f)

I am wondering how to describe the graph g in terms of the graph of f for these cases: $g(x)=f(1/x)$ $g(x)=|f(x)|$ $g(x)= f(|x|)$ $g(x)=\max(f,0)$ $g(x)=\min(f,0)$ $g(x)=\max(f,1)$
0
votes
2answers
28 views

How to prove a solution of equation is rational if another one is rational number?

The question is : $r$ is the solution of equation $x^2+bx+c=0$ and $r$ is a rational number, so there is another solution $s$, how to prove s is a rational number as well? I have no idea about it and ...
0
votes
0answers
19 views

Logarithmic function transformations

The standard log function form is $a \log[k(x-d)] + c$ Where $a$ vertically stretches or compresses $k$ horizontally stretches or compresses $d$ translates left or right $c$ translates up or ...
0
votes
2answers
47 views

Is a factorable polynomial invertible?

The reason there exists no quintic formula that finds the roots of a quintic polynomial is simply because some quintic polynomials are irreducible. But reducible quintic polynomials may be invertible ...
-1
votes
2answers
28 views

Basic: $1$ unit costs $\$10.00 $. Increases by $\$50 $ every unit. Total cost for $1000$ units? [on hold]

The cost for $1$ car is $\$10.00$. Every time you buy $1$, the cost increases by $\$50$. What is the cost for $1000$ units. If you have $10$ million dollars, how many units can you buy. Thanks Peter
0
votes
1answer
13 views

Prove that: $n^2+3n^3 + 6^{lgn} is $ $\theta(n^3)$

I'm asked to prove that: $n^2+3n^3 + 6^{lgn} is $ $\theta(n^3)$ I know that for Big O, I need to show: $f(n) <= c*g(n)$ But I'm not sure how to show this, since it involves theta. Any help would ...
0
votes
2answers
33 views

How to know the existence of solution of algebra equation?

For example we want to find a and b such that av+bw=0 (bold text means vector, otherwise scalar) Usually we would just solve the equation. But before solving that equation we need one assumption: ...
1
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0answers
33 views

Basic optimization question

A teacher put this problem up the other day and I'm confused about how he got to the answer. Can you explain it to me? Job $X$ provides $20$ vacation days and $143,000$ euro annual salary. Job $Y$ ...
1
vote
3answers
48 views

Solve nonlinear system of equations

Solve the system of equations $$\begin{cases}163-400z\sin{x}&=0\\-135z+85\cos{x}+61&=0\end{cases}$$ What is the best way of going about this? I rearranged the second equation for $z$ and ...
1
vote
2answers
62 views

how to prove this inequality $(ab+bc+ac)^2 ≥ 3abc(a+b+c)$

Prove that if $a,b,c$ are non-negative real numbers, then $(ab + bc + ca)^2 \geq 3abc(a+b+c)$. I tried to compute from $(a-b)^2 + (b-c)^2 + (c-a)^2 \geq 0$.
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votes
2answers
43 views

Simplify Expression if possible [on hold]

Good day all. Please help me to simplify the following expression if possible: $$x^n + x^m$$
0
votes
2answers
28 views

Find the inverse of a matrix with variable a ≠ 0

I have this matrix below and I'm trying to find it's inverse, I know I augment it with I2 but I don't know where to go from that. \begin{bmatrix} 2&1\\ a&a \end{bmatrix}
0
votes
0answers
25 views

Solve and equation for 2 variables

A bit of background: I'm writing a report in a piece of software and have to use XSLT1.0 (no extensions supported). What this means is that I only have basic arithmetic functions and a few simple ...
1
vote
2answers
42 views

Is this function bounded above?

Consider nonconstant functions $f(x), g(x) \neq x$. Suppose there exist positive constants $k_1$ and $k_2$ such that $k_{1} x \leq f(x) \leq k_{2} x$ and $\frac{1}{2}k_{1} x \leq g(x) \leq k_{2} x$. ...
0
votes
3answers
29 views

Find the condition such that one of the lines defined by $ax^2+2hxy+by^2=0$ has slope $k$ times that of the other

Find the condition that the lines represented by $$ax^2+2hxy+by^2=0$$ are such that the slope of one line is $k$ times that of the other. I calculated the two represented by $ax^2+2hxy+by^2=0$ ...
0
votes
3answers
53 views

how **(1)** $(2n-1)\pi/2 + (-1)^n\pi/3$ and **(2)** $2n\pi±\pi/6$ indicates the same angle?

I'm learning Trigonometry right now with myself and at current about General solution. I have a question in my book which I don't understand how to proof. The question is Show that the two angles are ...
2
votes
1answer
29 views

Inquiry on big $O$ notation

As a deeply enthusiastic prospective undergraduate student, there are is a fact that i'm still to completely understand about the big $O$ notation, namely: Let $f(x), g(x) \neq x$ be nonconstant ...
1
vote
0answers
33 views

Simultaneous equation with 6 equations, 6 unknows and a degre of 5

How do I solve this simultaneous $$a + b + c = 2\to (1)$$ $$ax + by + cz = 0\to (2)$$ $$ax^2 + by^2 + cz^2 = \frac{2}{3}\to (3)$$ $$ax^3 + by^3 + cz^3 = 0\to (4)$$ $$ax^4 + by^4 + cz^4 = ...
1
vote
3answers
70 views

How many blue rectangles there are?

Blue and red rectangles are drawn on a blackboard. Exactly 7 of the rectangles are squares. There are 3 red rectangles more than blue squares. There are 2 red squares more than blue rectangles. How ...
4
votes
1answer
28 views

Does $\sin^{-1}x$ has a vertical tangent

I read that the function $f(x)$ has a vertical tangent at $x=a$ in the domain of $f$ if $$f'(a^-) \to +\infty$$ and $$f'(a^+) \to +\infty$$ Or both approach to $-\infty$. But for $f(x)=\sin^{-1}x$ ...
-2
votes
2answers
81 views

Find $a$ and $b$ such that $\sqrt{9+6\sqrt{2}}=a+b$

I am trying to find $a$ and $b$ such that $\sqrt{9+6\sqrt{2}}=a+b$, obviously this can just be evaluated by a computer to find $\sqrt{3}$ and $\sqrt{6}$ but I'm wondering if there's a relatively ...
1
vote
1answer
13 views

Prove that the equation of the tangent at P is $ \frac {xx_1}{a^2} - \frac {yy_1}{b^2} = 1 $ (Hyperbolas)

Question: Point P ($x_1 , y_1$) is on the hyperbola $\frac {x^2}{a^2}$ - $\frac {y^2}{b^2}$ = 1 Prove that the equation of the tangent at P is $$ \frac {xx_1}{a^2} - \frac ...
4
votes
5answers
78 views

About Factorization

I have some issues understanding factorization. If I have the expression $x^{2}-x-7$ then (I was told like this) I can put this expression equal to zero and then find the solutions with the quadratic ...
0
votes
0answers
21 views

What is the maximum distance between the top of the door and the top of the window?

Problem: A rectangular window is to be cut into a door 6.5 ft tall so that the window is 6 inches less than twice the distance above the window. The window must be 18 inches wide and have an area of ...
0
votes
0answers
9 views

What function gives this inequality?

Let $i<j<k<l$ be positive integers. I want to find a "nice" function $f(x, y)$ such that $f(i, k)+f(j, l)>\max(f(i, j)+f(k, l), f(i, l)+f(j, k))$. This seems a bit tricky because the ...
0
votes
1answer
22 views

algebraic rearrangement, $C=h/[m(t_1-t_2)]$ Solve for $t_1$

$$ C=\frac{h}{m(t_1-t_2)} $$ Solve for $t_1$. The correct answer I have been given is, $t_1=t_2-h/(mC)$. I just need help in the steps taken to reach this.
0
votes
1answer
34 views

When will they meet up together? [on hold]

Jabal and Michael are walking to school and agree to leave at the same time. Jabal lives 100 meters closer to school. Jabal walks 2 meters per second. Michael walks 2.5 meters per second. When ...
1
vote
1answer
33 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 $\xi\in\mathbb{R}^d$. Does this somehow imply ...
0
votes
2answers
38 views

what is the n-k derivative of $x^n$? Also, why is $n!/k! = …$

I am having troubles finding $\frac{d^{n-k}x^n}{dx^{n-k}}$ where $ k \leq n$ I believe it is equal to $n(n-1)(n-2)....k(k+1)x^k$ but htis is just from obersation, I do not know why it's that exactly. ...
0
votes
3answers
62 views

For what values ​​of a intersects $y = ax$, $y = \sin x$ just one time?

As the title says, for what values of a intersects $y = ax$, $y = \sin x$ just one time? I am not able to solve this problem, and I really want to know the answer.
-3
votes
2answers
35 views

find the value of $k$ in the term $2^{-k} = 1/n$

What is the value of $k$ if I have the following equation: $2^{-k} = \frac1n$? $$2^{-k} = \frac 1 n \implies n = 2^k \implies \log_{2} n = k$$ Is my solution correct?
2
votes
3answers
63 views

$\sin2(x) - \tan(x) = 0$ , solve for $-180\le x\le 180$

I have been unable to solve the following question, If $$\sin(2x) - \tan(x) = 0$$ Find $x$ , $-\pi\le x\le \pi$ So far my workings have been Use following identity: $$\sin(2x) = ...
3
votes
4answers
58 views

Number of integer solutions of $\frac{1}{x}+\frac{1}{y}=\frac{1}{2016}$ [on hold]

How can we find number of integer solutions of $\frac{1}{x}+\frac{1}{y}=\frac{1}{2016}$ I want to ask what approach in general should be followed in such types of question?
1
vote
3answers
49 views

Which fraction is more

Can anyone help me prove why $$1-\left(\frac{x-10}{y}\right) \gt 1-\left(\frac{x}{y+10}\right)$$ when $y < x$. I have no idea how to show this algebraically so i'd really like some guidance.
3
votes
2answers
49 views

quadratic simultaneous equation

Solve simultaneously: $$ 12x^2-4xy+11y^2=64 $$ $$ 16x^2-9xy+11y^2=78$$ I understand that it can be solved using the quadratic formula by rearranging the equation in $ax^2+bx+c=0 $ form $$ ...
0
votes
1answer
31 views

Order of math operations

I know that this seems a childish question but i have not been able to find a proper convention that cover all math operations. Obviously the basic is that exponential takes precedence over ...
0
votes
1answer
46 views

Do we have $\frac{1}{a} - \frac{1}{b} = b - a$?

I am attempting to prove that $$\frac{1}{E'} - \frac{1}{E} = \frac{1}{m_e c^2} \cdot (1-\cos\theta)$$ can be derived from $$E + m_ec^2 - E' = c^2(p^2 - 2pp'\cos\theta + p'^2) + m_e^2c^4 $$ where ...
1
vote
1answer
24 views

Supremum of integral polynomial near origin

Let $P(x,y)$ be a polynomial with integer coefficients that is constant neither in the horizontal nor vertical direction. Prove that $\sup_{-2\leq x,y\leq 2}|P(x,y)|\geq 4$. I suspect we might be able ...
0
votes
1answer
18 views

How to use Rolle's theorem to verify the following?

How to use Rolle's theorem to verify the location of roots ? $f(x)=x^3+4/x^2+7$ has exactly one zero in ($-\infty$,$0$) I can do it without Rolle's theorem by finding the stationary point which is ...