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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0answers
16 views

Existence of solution for an equation including polynomial and trinogometric sum

Prove that the following equation has at least a solution in $[-\pi, \pi]$ : $$ x^5+\sum^{n}_{k=1}(a_k\cos kx+b_k\sin kx)=0 $$ I think the existence of the solution on $[-\pi, \pi]$ strongly depends ...
1
vote
1answer
20 views

Maximum and minimum of a fractional function

Let $x, y \in \mathbb{R}$, $a, b, c$ are three real parameters with $c\neq 0$. Find the maximum and minimum of $\dfrac{ax+by+c}{\sqrt{x^2+y^2+1}}$ This is quite complicated if I calculate the ...
3
votes
1answer
74 views

$\sqrt{m_1}+\sqrt{m_2}+ \cdots + \sqrt{m_n}$ is Irrational

If $m_1 , m_2, \cdots m_n$ are natural numbers where at least one of them is not a perfect square, then how do I prove that the sum $$\sqrt{m_1}+\sqrt{m_2}+ \cdots + \sqrt{m_n}$$ is irrational? I'm ...
0
votes
2answers
30 views

variables to the power of a fraction

I have this question for advanced math, I can't seem to get my head around. $$\frac{x^{5/2}}{(x^{1/3})^4}$$
0
votes
7answers
652 views

Is it possible to find the product of two numbers given their difference?

That is, find $a\cdot b$ given the value of $a-b$. Is it possible?
0
votes
1answer
38 views

How to solve quartic polynomial equation

Can someone tell me how to solve $x^4 + 6x^2 + 5 = 0$? I know what to do when each term has an exponent one less than the previous term (e.g., $x^4 + 3x^3 + 6x^2 + 5 = 0$), but not when exponents are ...
0
votes
0answers
10 views

Displaying a 3D function without a graph

I have a 3D function $z=\dfrac{x}{y}$, and I have no access to a function grapher, but I still need to display this function in a comprehensible way. I thought of a table, but even with a domain of ...
-1
votes
0answers
17 views

Combing piecewise functions [on hold]

How would I combine the following two piecewise functions in terms of addition and subtraction? How would I find $f(x) + g(x)$, and also $f(x) - g(x)$? Thanks!
0
votes
0answers
12 views

Iteratively solve this equation

I am supposed to solve $1 = \left( \frac{\mu}{f} \right)^{\frac{3}{2}} \left( 1+ \frac{ \pi^2}{8} \left( \frac{kT}{\mu} \right)^2 \right)$ iteratively for $\mu$ and am supposed to get $$\mu = f ...
15
votes
7answers
943 views

How to make a “function”?

I dropped out of school early when I was still a teenager and now I'm trying to take my GED. I'm really close to passing but I'm still having trouble understanding some concepts. In the pre-test, ...
0
votes
1answer
15 views

Exclude one function from another

Is it possible to find a function, $g(n)$ that will include all the natural values except those in $f(n)$? $$f(n) = 3n$$ $$g(n) = 1,2,4,5,7,8,10...$$
0
votes
2answers
23 views

Line Intersect in Diagonals of a Rectangle

The diagonals of the rectangle have these equations: $$y = 4x-10\\ \\ y = -4x+18$$ Find the point at which the diagonals intersect. First, I tried working out $(x,y)$ $4x - 10 = -4x + 18$ $4x = ...
0
votes
0answers
8 views

Bound all $k$-th derivatives by directional derivatives of order $k$

Assume $f\in C^k(\mathbb{R}^n)$, $x\in\mathbb{R}^n$, and $|(\partial_\xi)^kf(x)|\leq 1$ for all $\|\xi\|=1$. Which bounds do we have for $|\partial^\alpha f(x)|$ when $|\alpha|=k$? For example, if ...
3
votes
4answers
477 views

Given that a,b,c are distinct positive real numbers, prove that (a + b +c)( 1/a + 1/b + 1/c)>9

Given that $a,b,c$ are distinct positive real numbers, prove that $(a + b +c)\big( \frac1{a}+ \frac1{b} + \frac1{c}\big)>9$ This is how I tried doing it: Let $p= a + b + c,$ and $q=\frac1{a}+ ...
1
vote
0answers
71 views

Crossed Ladders Problem

Two ladders, one 10 meters long and the other 8 meters [long], have been placed in a trench as indicated in the opposite figure. Their point of intersection, M, is 3 meters from the base of the ...
-1
votes
0answers
17 views

Which of the Following statements are true? algebra 2 [on hold]

I need help with this problem. Help me find out which one of the statements are true. There can be more than one. I'm positive that one of them is A.
0
votes
2answers
28 views

General Formula for Principle Square Root of Complex Number

How can I prove that $ \sqrt{z} = \sqrt{|z|} \frac{(z + |z|)}{|z+|z||} $ without using mathematical induction, and if I cannot -- how would I go about using induction in the set of complex numbers ?
0
votes
0answers
28 views

Logarithms: expanding, condensing, inverse, and checking for extraneous solutions. [on hold]

They're all separate so one answer doesn't apply to others, but I need help with how to condense logarithms, find the inverse, and check for extraneous solutions. First I'm condensing logarithms, ...
0
votes
2answers
35 views

Show$\:\frac{1}{\left|x^2+x+1\right|}\:\ge \:\frac{1}{x^2-\left|x\right|-1}$

This is the answer I can come up with. I get the complete opposite of what I'm supposed to get. My mistake is probably in the first part, could anyone help me out? $$\left|x^2+x+1\right|\:\ge ...
-5
votes
3answers
38 views

The lines with equations $y = 5x − 6$ and $10x + cy = 8$ are perpendicular, find c [on hold]

The lines with equations $y = 5x − 6$ and $10x + cy = 8$ are perpendicular. Find the value of c. Well, I am not sure even where to start
0
votes
1answer
49 views

$(a+\frac{1}{2})^n + (b+ \frac{1}{2})^n$ is an integer for at most finitely many $n$ [duplicate]

Prove that for any positive integers $a,b$ $(a+\frac{1}{2})^n + (b+ \frac{1}{2})^n$ is an integer for at most finite number of integers $n$. Here is what I tried ; I tried to use mathematical ...
-4
votes
1answer
35 views

Algebra,complex numbers home work problem [on hold]

Please I want the solution of this problem : $z= \dfrac{(2-i) \cdot (x+4i)}{3-4i}$ and $|z|=2$ then $X=?$
0
votes
5answers
49 views

Number theory proof [on hold]

$(i)$ Prove that for every natural number $n \geq 2$, one has $(n + 1)|(n^3 + 1)$; $(ii)$ Suppose that $n$ is a natural number exceeding $1$. Prove that $(n^2-1)|(n^3+1)$ if and only if $n = 2$.
4
votes
0answers
46 views

Rationality and triangles

Consider a triangle with angles $\alpha, 5\alpha, 180-6\alpha$. What is the minimum perimeter of that triangle, if it has integer sides and $5\alpha<90$?. Let's call tha sides that face each ...
0
votes
1answer
52 views

what is going on here?

Suppose we have a function $f(x), D:( -\infty,0)\cup (0,\infty)$ and for which $$f'(x) = \frac{x^3-1}{x^3} $$ Apparently there is only one point of extremum here, $x=1$, however upon reviewing the ...
0
votes
1answer
47 views

Polynomials, prove exercise question about question

There is a polynomial P with integer coefficients and with pairwise different integers $a,b,c$ . Prove that it is not possible for $P(a) = b$, $P(b)=c$, $P(c) = a$ First off I don't understand ...
1
vote
1answer
24 views

Finding an Expression for the Difference of Roots of the Quadratic Equation

Let the equation $ax^2+bx+c=0$ have the roots $\alpha$ and $\beta$, then what is $\alpha-\beta$ in terms of $a$, $b$, and $c$? Well, we may write $$(\alpha-\beta)^2=(\alpha+\beta)^2 -4\alpha \beta$$ ...
0
votes
1answer
14 views

solve $k(k-1) \geq \ln2*2m$ for k

My Question is related to the birthday problem. Starting at $e^{-\frac{k(k-1)}{2m}} \leq 0.5$ i used $ln(x)$ on both sides and multiplied by $-2m$ to get $k(k-1) \geq \ln2*2m$ According to my ...
-1
votes
0answers
29 views

Slope of the secant line given a point

The point $P(1,0)$ lies on the curve $y= \sin(10\pi/x)$. (a) If $Q$ is the point $(x, \sin(10\pi/x))$, find the slope of the secant line $PQ$ (correct to four decimal places) for $x=2, 1.5, 1.4, 1.3, ...
1
vote
0answers
28 views

How can I solve this para Paradox? [duplicate]

How can I solve this para Paradox? $ -1={(-1)}^{1/2} {(-1)}^{1/2}={[(-1)(-1)]}^{1/2}=1$
-7
votes
4answers
94 views

Can $9+10$ equal $21$? [duplicate]

I just saw a video with the following: $9+10=21$ because $$9 = 3*3$$ $$10 = 5*2$$ if $$5*3=15$$ and $$2*3=6$$ $$15+6=21$$ does that not prove that $10+9=21$?
0
votes
1answer
29 views

Which functions are power functions? [on hold]

I have 6 functions and want to know which ones are power functions and which ones aren't. I know that the power function has the form $f(x)=Kx^p$, where $K$ and $p$ are constants. $f(x)= \pi*x^4$ ...
-1
votes
2answers
19 views

Find closed form for a sequence using the Fibonacci and Lucas number sequences. [on hold]

Define $A_n$ as follows: $$\begin{align*} A_0 &= 6 \\ A_1 &= 5 \\ A_n &= A_{n - 1} + A_{n - 2} \; \textrm{for} \; n \geq 2. \end{align*}$$ There is a unique ordered pair $(c,d)$ such that ...
-11
votes
0answers
59 views

Solving equation please help [on hold]

Solve for $x$ $$-6(x-7)=-2(4+3x) $$
4
votes
2answers
101 views

How do I simplify this expression about factorization?

I am trying to simplify this $$\frac{9x^2 - x^4} {x^2 - 6x +9}$$ The solution is $$\frac{-x^2(x +3)}{x-3} = \frac{-x^3 - 3x^2}{x-3} $$ I have done $$\frac{x^2(9-x^2)}{(x-3)(x-3)} = ...
1
vote
1answer
18 views

Creating a weighted score

I have an audit where there are six criteria, each can be scored Excellent (E), Satisfactory (S), Needs improvement (N) or Unsatisfactory (U). I know that if someone scores Excellent in all six areas ...
0
votes
2answers
55 views

Matrix invertible iff det(matrix)$\neq 0$?

When we want to find the inverse of the matrix $$\begin{bmatrix}a & b \\ c & d\end{bmatrix}$$ we're searching for a matrix $$\begin{bmatrix}x & y \\ z & w\end{bmatrix}$$ such ...
2
votes
1answer
36 views

Differents between $lnx^2$ and $ln(x^2)$ Find derivative

I have this problem Find derivative for $lnx^2$. It seems that $lnx^2 \neq ln(x^2)$ since the derivative are differents using Wolfram Alpha. I don't understand how to calculate the derivative for ...
2
votes
7answers
638 views

Working Out Easy Equations

does anyone know how to do this equation? I know it's easy but I can't work out what the question means. When I expanded the first equation: $(y+4)-(y-3)$ $y^2 -3y +4y - 12$ $y^2-1y-12$ Not ...
0
votes
1answer
40 views

Getting ready for Calculus?

So I wanted to start a Masters program but they require that I have Calculus III. I want to take that course at the university, but I need to be ready for it. As I look at Khan Academy and do some ...
2
votes
1answer
30 views

Calculating $\sum_{k=0}^{n-1}\frac{1}{a+bk^2}$.

I want to calculate the following summation: $$\sum_{k=0}^{n-1}\frac{1}{a+bk^2}$$ Any hint how I can calculate this? Is there any kind of closed form for this summation?
2
votes
1answer
30 views

Algebraic values of the sine function

First question: For which angles $x$ is $\sin(x)$ a real number that can be expressed using only integers, addition, subtraction, multiplication, division and the extraction of $n$th roots? (With ...
2
votes
1answer
30 views

Inequality $(a+b)^2 + (a+b+4c)^2\ge \frac{kabc}{a+b+c}$ for $a,b,c \in\mathbb{R}$

Find biggest constans k such that $(a+b)^2 + (a+b+4c)^2\ge \frac{kabc}{a+b+c}$ is true for any $a,b,c \in\mathbb{R}$ Could you check up my solution? I'm not sure it's ok - $(a+b)^2 + (a+b+4c)^2 \ge ...
2
votes
1answer
48 views

Inequality $x^4+y^4+(x^2+1)(y^2+1)\ge x^3(1+y) +y^3(1+x)+x+y$ for $x,y \in\mathbb{R}$

Prove for $x,y \in\mathbb{R}$ that such inequality exists ; $x^4+y^4+(x^2+1)(y^2+1)\ge x^3(1+y) +y^3(1+x)+x+y$ And here is what I realised ; because $(x^2+1)(y^2+1) >=1$ and $x^4+y^4 \ge 0$ ...
-2
votes
4answers
66 views

How to determine the nth term of a sequence, given only the first four terms? [on hold]

I need to calculate the tenth term of the following sequence: $1 \quad 8 \quad 27 \quad 64\quad \ldots$
3
votes
1answer
49 views

Solving A Certain Diophantine Equation

I am stack on finding the solution of the diophantine equation: $d(2^{k+1}-1)-b^2(2^{k+1}-2)=1$. where $k\geq 1$ and $b^2>d$ for $b$ an odd composite integer. Is there a solution to this ...
-4
votes
1answer
34 views

Linear-algebra problem [on hold]

How do you solve this equation? $$ -7x-28=7+6(1-8x)$$
0
votes
1answer
47 views

How can the modulus of something be less than zero?

I've been asked to prove that for $\epsilon>0$, $$| a-x| < \epsilon \iff a-\epsilon<x<a+\epsilon,$$ and as a hint to consider both $| x-a|>0$ and $| a-x|<0$. I used the fact that ...
2
votes
2answers
60 views

How many positive integers less than $2013$ are divisible by none of $2, 3, 4 ,5$?

How many positive integers less than $2013$ are divisible by none of $2, 3, 4 ,5$? This was an olympiad question. I thought of writing a number $x \le 2012$ in the form: $x = 2^{a}3^{b}4^{c}5^{d} = ...
0
votes
2answers
14 views

FV (Future Value) of annual payments

My client will receive $\$881$ now and twice that amount in a year. He will get 3 times $\$881$ in 2 years . 4 x $\$881$ in 3 years etc. for as long as he lives. Assuming he lives 20 years and each ...