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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2
votes
0answers
28 views

Trigonometrical functions and complex numbers

(This question will at first appear too broad. However, the overall philosophy will be explained below in a way that asks specific questions, which I hope will be conducive to this being a reasonable ...
0
votes
2answers
42 views

How does $ \left({(n+1)}^4 \over 4^{n+1}\right)\left(4^n \over n^4 \right) $ simplify?

$$ \left({(n+1)}^4 \over 4^{n+1}\right)\left(4^n \over n^4 \right) $$ Just not sure how this is able to simplify any more.
1
vote
0answers
41 views

Is this simplification 'allowed'?

I've just been doing a problem that involved this equation: $$ \frac{1}{\sin\left(\frac{\theta}{2}\right)}\left( \sin\left(b\theta-\frac{\theta}{2}\right)-\sin\left(a\theta-\frac{\theta}{2}\right) ...
4
votes
1answer
25 views

Precalculus equation solving involving square roots

Solve the equation: $$\sqrt x = \frac{3}{\sqrt x}+ \sqrt {x+3}$$ My approach was to multiply both sides with $\sqrt x$: $$x = 3 + \sqrt {x+3} \sqrt x$$ $$(x - 3)^2 = (x+3)x$$ $$x^2 - 6x + 9 = 3x ...
0
votes
0answers
13 views

respective values of two functions are “closer than expected”

let f,g be functions from the same finite set into the reals, let d be the mean distance between f(x) and g(x) for x in S, and let D be the mean distance between f(x) and g(y) for x,y in S; then D-d ...
-2
votes
1answer
22 views

question from real numbers [on hold]

a man has 1044 candels after burning he can make a new candal from 9 stubs left behind .the maximum numbers of candals that can be made are?
-1
votes
4answers
32 views

Square in the complex plane given three vertices. Find the fourth complex number vertice.

There is a square in the complex plane. Four complex numbers form the four vertices of this square. Three of the complex numbers are $-19 + 32i,$ $-5 + 12i,$ and $-22 + 15i$. Find the fourth complex ...
0
votes
0answers
26 views

$(1+\cos x + 2\sin x)\dfrac{dy}{dx} + (\sin x-2\cos x)y = 5-3\cos x + 4\sin x$ case when $1+\cos x+2\sin x = 0$?

Find the general solution of $(1+\cos x +2\sin x)\dfrac{dy}{dx} + (\sin x-2\cos x)y = 5-3\cos x + 4\sin x$ I solved it, with dividing both sides by $(1+\cos x+2\sin x)^2$, Solution giving ...
9
votes
7answers
899 views

Inequality with (1-x) as denominator

How do I solve $\frac{1}{x-1}>0$ for $x$? If I multiply both sides with $x-1$ then becomes $1\gt 0$. I know it's wrong. How do I solve it?
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 ...
-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
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. ...
0
votes
1answer
13 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 ...
-1
votes
1answer
9 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 ...
-3
votes
2answers
43 views

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

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. ...
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
1answer
33 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
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 ...
-2
votes
1answer
25 views

Write a polynomial with the following zeros: -2 multiplicity of 1 and 0 with a multiplicity of 2. [on hold]

I am unsure about how to complete this problem. Will the solution be factors? ex: (x+1)(x+2)
1
vote
2answers
56 views

Given $r>0$, find $k>0$ such that $\sqrt{(x-2)^2+(y-1)^2}<k$ implies $|xy-2|<r $

Using the axioms, theorem, definitions of high school algebra concerning the real numbers, then prove the following: Given $r>0$, find a $k>0$ such that: $$\text{for all }x, y: ...
1
vote
2answers
34 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 ...
-5
votes
2answers
30 views

Ticket price word problem - Simultaneous equations [on hold]

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 ...
-1
votes
1answer
34 views

Exponential Function Equation and inverse Pre-Cal

I am not completely sure if I wrote the equation correctly. For A I wrote: $m(t)=100(b^x)$ Not sure it is correct...but how do I find the inverse? That doesn't make sense to me. Do I use log?
1
vote
4answers
45 views

Logarithm Question (Find x) [on hold]

How to solve x for $$x^{2\log_{10}x}=\frac{x^5}{100}$$?
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 = $P(2)$ = ?
0
votes
2answers
29 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 ...
4
votes
5answers
681 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 ...
-2
votes
1answer
36 views

Find minimal $x$ and $y$ that creates $4$ [on hold]

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
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 ...
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 ...
-2
votes
2answers
29 views

Use the fundamental identities to simplify the expression. [on hold]

Use the fundamental identities to simplify the expression $$ 4 \sin x (\csc x - \sin x) $$
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 ...
3
votes
3answers
65 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, ...
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.
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
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 ...
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, $r+s+x_1$ $=0$ ...
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 ...
-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 $\alpha^2-\beta^2$.
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 $t$ ...
-1
votes
4answers
54 views

Multiplying whole number with fractions.

I'm looking at a solution to a math problem and there are obviously some rules regarding multiplication of fractions that I don't know. Can someone make any sense of this? $$s_n = 625 \cdot ...
1
vote
2answers
23 views

Absolute Value Algebra with inverses

I noticed the following equality in some material regarding limits and infinite series. $$ \left |\frac{x}{x+1} - 1 \right| = \left |\frac{-1}{x+1} \right| $$ And I'm honestly stumped (and slightly ...
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 ...
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
1answer
38 views

Find the volume of the region bounded by $ (x^{2}+y^{2}+z^{2})^{2}=x$

I tried to convert it to spherical coordinates to find the bounds: $(p^{2})^{2} = p\sin(\phi) \cos(\theta)$ => $ p^{3} = \sin(\phi)\cos(\theta)$ not sure where to go from here.. $ 0 < \theta ...
0
votes
3answers
31 views

Algebra rearranging

I'm stuck on a question that should be extremely easy. The idea is to show that $$\frac{\frac{n}{z}y + a}{y + \frac{z}{n}a} = \frac{n}{z}$$ What is the best method here? Thanks