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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-5
votes
3answers
32 views

problem involving percentage [on hold]

If 95% of the students are present and number of absent students are 25.how many total number of studrnts are there?
0
votes
0answers
15 views

Solve a high order polynomial equation in $x$ in the limit $n\rightarrow\infty$

A bit of background. I did a high order WKB theory to calculate the eigenvalues of a potential. The eigenvalues, $E$, are, of course, real since they correspond to a physical problem. My final answer ...
0
votes
1answer
35 views

Reversing equation with a logarithm and exponent

This is my equation: $$ x=y^{3.333+(-1(0.5\times \log_{10}(10-y)))} $$ It will solve for x, with input of any y. I want to solve for y with input of any x.
0
votes
1answer
14 views

natural logarithmic to asymptotic order

Say we have an equation $\lambda_{\epsilon}(s)=-\frac{1}{\pi s^2}\ln(1-\epsilon)$ $\forall s\in (0,(M \mathcal{k})^{-\frac{1}{\alpha}})$ where $s$ can be obtained by $s=(M ...
0
votes
2answers
21 views

If $a_n=n^x(n^{1/n^2}−1)$, show that $\ln{(1+a_n/n^x)} = \frac{\ln(n)}{n^{2}}$

Let $$a_n=n^x(n^{1/n^2}−1).$$ Show that $$\ln{(1+a_n/n^x)} = \frac{\ln(n)}{n^{2}}. $$ It is on the study guide for my final exam, which is tomorrow so I am trying to figure it out. ...
-2
votes
0answers
27 views

Calendar question [closed]

A certain day, which is $x$ days before $17$th of August, is such that $50$ days prior to that day, it was $4x$ days since March $40$th of the same year. What is $x$?
-1
votes
1answer
11 views

Scaling the subtraction of volume [on hold]

Phil is making a fruit drink. He has one large jar filled with $96$oz of water. However, this is too much water, and he needs to get rid of some to make room for other ingredients. He removes $25$oz ...
1
vote
4answers
45 views
-3
votes
2answers
44 views

$x^2 +x -2 \geq 0$ basic problem [closed]

if $x^2 +x -2 \geq 0$ How do I conclude $x \geq 1$ or $x \leq -2$ algebraically ?
0
votes
1answer
37 views

Can someone help me with this Partial Fraction Decomposition?

Can someone help explain the steps to solving this problem? I can't seem to figure out how to find the answer when the denominator is unfactorable or has a cubed exponent for the nominator. ...
0
votes
1answer
379 views

Irrational inequalities question: $\sqrt { -3x+1 } + \sqrt {6x+1} < \sqrt {3x+4}$ and $\sqrt { -6x+10 } + \sqrt {-x+2} \gt \sqrt {4x+5}$

Consider the following inequalities: $\sqrt { -3x+1 } + \sqrt {6x+1} \lt \sqrt {3x+4}$ $\sqrt { -6x+10 } + \sqrt {-x+2} \gt \sqrt {4x+5}$ Attempt at a solution; after performing all the ...
1
vote
4answers
2k views

Smooth transition between two lines (2d)

I have function that is defined as $$ Y = \frac{1}{15} x \longrightarrow {\rm if}\qquad 0 \leq x \leq 30 $$ $$ Y = \frac{1}{70} x + \frac{11}{7} \longrightarrow {\rm if}\qquad x > 30 $$ The ...
1
vote
2answers
21 views

Confusion about division in rates.

I would really appreciate help with this because it's been driving me insane for a while now... I understand what "per" means in "$x$ kilometers per $y$ hours". What I don't understand is how to make ...
0
votes
2answers
23 views

Is it possible to have a system of equations that all equal 0, and not have each unknown's value be 0?

I'm doing about a 2 hour long homework assignment where by hand I must construct a 10x10 matrix representing a system of equations. Based on the pattern I'm seeing, I can tell all of the equations ...
1
vote
2answers
36 views

Prove equality of two numbers written in complex polar form.

Show that these two numbers are equal: $$ z_1=\frac{e^{\tfrac{2\pi i}{9}}-e^{\tfrac{5\pi i}{9}}}{1-e^{\tfrac{7\pi i}{9}}} $$ and $$z_2=\frac{e^{\tfrac{\pi i}{9}}-e^{\tfrac{3\pi ...
18
votes
7answers
2k views

Why aren't these negative numbers solutions for radical equations?

I was working on radical equations and I came across a few problems where I got answers that worked when I checked, but were not listed as solutions. My teacher's only explanation was, "just because." ...
-2
votes
1answer
25 views
-5
votes
2answers
30 views

Ticket price word problem - Simultaneous equations [closed]

Jen has been pricing speed-pass train fares for a group trip to NY. Three adults and four children must pay $\$101$. Two adults and three children must pay $\$71$. Find the price of the adults ticket ...
0
votes
1answer
28 views

Use the remainder theorem to find $P(2)$ where $P(x)=-x^4+3x^3-4x+7$

Use the remainder theorem to find $P(2)$ where $P(x)=-x^4+3x^3-4x+7$ Quotient = ? Remainder = $P(2)$ = ?
4
votes
5answers
683 views

My dilemma about $0^0$ [duplicate]

We know that $0^0$ is indeterminate. But if do this: $$(1+x)^n=(0+(1+x))^n=C(n,0)\cdot ((0)^0)((1+x)^n) + \cdots$$ we get $$(1+x)^n=(0^0)\cdot(1+x)^n$$ So, $0^0$ must be equal to $1$. What is ...
0
votes
2answers
30 views

Solution set of inequality

This is the question: $$\frac{1-2x-3x^2}{3x-x^2-5} \gt 0$$ What I did : I got the answer as $$\left(x-3\right)\left(x+1\right) \gt 0$$ giving me the solution set : $x \in (-\infty,-1 ...
0
votes
1answer
34 views

How to solve over-determined linear system of equations?

at the moment I am working on an application where I have to solve some systems of linear equations during the whole algorithm. Because the programming-language I have to use is something related to ...
0
votes
0answers
19 views

Limit of solution equal to solution at limit

I have a system of equations (not DE, think algebraic equations) that depends on a parameter. I am trying to learn under what conditions on the system is a limit (as the parameter converges to some ...
2
votes
1answer
46 views

Two variable equation

I'm stuck with the following example (42.). Some help is much appreciated. Thank you.
0
votes
0answers
33 views

Can someone explain what independent linear equations are? [on hold]

Can someone explain what independent linear equations are? Specifically whether the following equations are independent, or even linear equations? $$\frac Y{X-1}=\frac XY$$ $$Y=\left(\frac ...
-1
votes
0answers
21 views

Number of escalator steps we can see [closed]

A man walks up an escalator that moves up and counts 50 steps. The next day he walks up the same escalator and counts 75 steps. If the second speed (in steps per time unit) is three times the first ...
4
votes
2answers
112 views

Evaluate $\lim_{n \to \infty} \int_{0}^1 \frac{n+1}{2^{n+1}} \left(\frac{(t+1)^{n+1}-(1-t)^{n+1}}{t}\right) \mathrm{d}t$

Evaluate $$\lim_{n \to \infty} \int_{0}^1 \frac{n+1}{2^{n+1}} \left(\frac{(t+1)^{n+1}-(1-t)^{n+1}}{t}\right) \mathrm{d}t$$ For this integral, I have tried using integration by parts and then ...
-2
votes
1answer
36 views

Find minimal $x$ and $y$ that creates $4$ [closed]

Hello I had this in my exam, I've never studied this and I am interested in knowing how to solve it, and what is the category of this type of question: For all positive numbers $x$ and $y$ such that ...
0
votes
1answer
67 views

Simplify $(x_1+x_2+\dots+x_m)^p$? [duplicate]

Is there a way to simplify $(x_1+x_2+\dots+x_m)^p$? Thank you!
0
votes
0answers
17 views

Expansion $(x_1+x_2+\dots+x_m)^p\,(y_1+y_2+\dots+y_n)^q$?

Based on multinomial series, we have $(x_1 + x_2 + \cdots + x_m)^p = \sum_\limits{k_1+k_2+\cdots+k_m=p} \frac{p!}{k_1!\, k_2! \cdots k_m!} \prod_\limits{1\le t\le m}x_{t}^{k_{t}}\,$. So what is ...
1
vote
2answers
50 views

$y=\cos(m \arcsin x)$ Validity of solution $\dfrac {dy} {dx}$ when $x=0$?

$y=\cos(m \arcsin x)$, for $ -1 < x < 1$ I want to find the value of $\dfrac {dy} {dx}$ when $x=0$ using the following way: $=> \arccos y = m\arcsin x$ $=> - \dfrac {1} {\sqrt {1-y^2}} ...
0
votes
0answers
12 views

are $g$, $f$ terms of equation of a circle different from that in the general equation of second degree?

I know the conditions for a general equation of second degree $$ax^2+by^2+2hxy+2gx+2fy+c=0$$ to be a circle are 1. $h=0$ 2. $a=b$ So following the conditions the equation becomes ...
1
vote
1answer
19 views

Formula to map a variable to another?

For example, I have a variable $x$ that contains the value $100$, and assume I also have a variable $y$ that contains the value $300$ is there a method to decrement $x$ by some amount and have $y$ be ...
3
votes
3answers
66 views

How various properties of numbers, operations are found?

I know that how the term "property" is defined. Definition: An attribute, quality, or characteristic of something. Like one of the property of addition is "commutativity" which behaves like, ...
2
votes
1answer
29 views

If $(a-1)(x^4+x^2+1)+(a+1)(x^2+x+1)^2 = 0$ are real and distinct, Then set of all values of $a$

If the two roots of the equation $(a-1)(x^4+x^2+1)+(a+1)(x^2+x+1)^2 = 0$ are real and distinct, Then the set of all values of $a$ is. $\bf{Options::}$ $(a)\;\; \displaystyle ...
3
votes
3answers
72 views

How to scale a random integer in $[A,B]$ and produce a random integer in $[C,D]$

I know there are many methods to scale a number from range $[A,B]$ to a range $[C,D]$, and I've searched over and over the web. I've seen this math.SE thread. I need to scale a big number (signed ...
-2
votes
2answers
29 views

Use the fundamental identities to simplify the expression. [closed]

Use the fundamental identities to simplify the expression $$ 4 \sin x (\csc x - \sin x) $$
3
votes
1answer
42 views

Finding all possible pairs of positive integer values

The ratio of the sum of two positive integers to their difference is $7:5$. If the the sum of the two numbers is at most $25$, find all possible values for the pair of numbers. Let $m$ be the first ...
0
votes
0answers
30 views

Help understanding exponential formula

I am reading a paper in which a group is approximating data that fits an exponentially declining curve. They use the following formula to fit the data, where τ is the y-axis variable and v is the x ...
1
vote
1answer
35 views

Real Numbers are Roots $r, s$.

Real numbers $r$ and $s$ are roots of $p(x)=x^3+ax+b$, and $r+4$ and $s-3$ are roots of $q(x)=x^3+ax+b+240$. Find the sum of all possible values of $|b|$. Using Vieta's Formulas, $r+s+x_1$ $=0$ ...
0
votes
1answer
42 views

Algebra question leading to a 3rd order equation solving.Any other answers?

if : $x+y+z=2$ , $ x^2+y^2+z^2=3$ , $xyz=4$ Then evaluate: $\frac {1} {xy+z-1} + \frac {1} {yz+x-1} + \frac {1} {zx+y-1}$ My try: $(x+y+z)^2=x^2+y^2+z^2+2(xy+yz+xz)=4 \rightarrow 3+2(xy+yz+xz)=4 ...
0
votes
3answers
38 views

Algebraic Manipulation

What is the best method to get the LHS equal to RHS? $\frac{n(n+1)(n+2)}{3} + (n+1)(n+2) = \frac{(n+1)(n+2)(n+3)}{3}$ Thank you.
1
vote
4answers
63 views

Proving by induction that $n^2 - 7n - 2$ is divisible by $2$

Now proving by induction is fairly simple. However, this is a multiple choice problem whose answers don't make any sense to me. The actual problem goes as follows: To prove by induction that $n^2 - ...
1
vote
2answers
18 views

The sum of the abscissae of the intersections of a cubic and a line

I remember being told in passing in a talk once the following theorem: Let $y=x^3$, and let $x_1,x_2,x_3$ be the abscissae ($x$ co-ordinates) of three distinct points on this cubic. Then ...
3
votes
3answers
32 views

Make $kt^2+(3k+1)t+4k+1$ constant?

Find $k$ such that $kt^2+(3k+1)t+4k+1=0$ is an identity (i.e. true for all $t$). E.g. $k=t+1$ doesn't work since you end up with a third degree polynomial in $t$ which determines $t$, making $t$ ...
8
votes
4answers
92 views

Sum of $2$ equal squares also a square

Is there an integer solution to $a^2 + a^2 = b^2$? Because there's this universift that has this logo of the pytagorean theorem where the two squares are equal, but I don't think it's possible.
-5
votes
0answers
31 views

Question about alpha and beta [closed]

If $\alpha$ and $\beta$ are zeroes of polynomial $4x^2-3x+8$. find the value of $\alpha^2-\beta^2$.
2
votes
1answer
57 views

Number of algebraic solutions to a formula related to a square tiling problem

How can many different sets of prime-factors fit together so well in this formula? I am curious about the number of solutions to the following equation: $$ r_3 = \sqrt{2}\; \frac{ 1 + r_1 (r_2 ...
2
votes
2answers
92 views

How to solve these simultaneous equations: $y=10x-3$ and $y=x^2-3x$?

Okay so I am normally good at these kinds of things but I received this problem that even the top people in my class had trouble solving. The problem is that everyone is getting different results. We ...
2
votes
4answers
36 views

Simultaneous Quadratic Equations: $x^2 + y ^ 2 - 2 x + 6y - 35 = 0$ and $2x + 3y = 5$

I've been given the task to simultaneously solve: $$x^2 + y ^ 2 - 2 x + 6y - 35 = 0$$ $$2x + 3y = 5$$ I've tried applying the substitution method by reordering the second equation to both $x$ and ...