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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2answers
3k views

Solving an inequality, the equality is facing the wrong way?

I'm suppose to solve a problem that goes like this. The graph for the following function f given by $f(x) = 115.82 \cdot 0.94^x + 5$, with $x \geq 5$, gives the temperature of the water after ...
1
vote
5answers
573 views

Square Roots of Complex Number $3-4i$

What I did $z^2=3-4i$ $(a+bi)^2 = 3-4i$ $a^2-b^2+2abi = 3-4i$ Then got 2 simultaneous equations $a^2-b^2=3$ and $2ab=-4$ Solve for $a^2$ in 1st equation: $a^2=3+b^2$ Subbed into 2nd equation ...
1
vote
4answers
129 views

Where exactly I am going wrong here?

A can of juice was $80\%$ full. $80\%$ of the contents were emptied into a glass and $81$ ml of juice was added to the can. Then the can became full to the brim. What is the capacity of the can ...
2
votes
3answers
2k views

On the Origin and Precise Definition of the Term 'Surd'

So, in the course of last week's class work, I ran across the Maple function surd() that takes the real part of an nth root. However, conversation with my professor ...
0
votes
2answers
101 views

Calculate Points for a Parallel Line

Given a line running through p1:(x1,y1) and p2:(x2,y2), I need to calculate two points such that a new parallel line 20 pixels away from the given line runs through the two new points. Edit: The ...
1
vote
1answer
90 views

Solution of an Equation

I have this equation, and I want to find solution for x. $\begin{align*}&(-2 x+2 α+1/(2 σ^2))\exp[(-(x-α)^2+(x-μ)/(2 σ^2))]+\\&(-2 x+2 β+1/(2σ^2))\exp[(-(x-β)^2+(x-μ)/(2 σ^2))]=0\end{align*}$ ...
2
votes
1answer
189 views

Recipe for solving equations

I am making a script that's solving algebra. I am 16 years old so my script should be able to solve all middle school equations. (Because this is the level I have when it comes to maths. Got the ...
42
votes
16answers
51k views

What is a real world application of polynomial factoring?

The wife and I are sitting here on a Saturday night doing some algebra homework. We are factoring polynomials and we both had the same thought at the same time: when are we going to use this? I feel ...
0
votes
1answer
50 views

Equations and Solving Them

I am trying to answer some problems regarding simple equations and rational numbers, and I have an algebra book, but it shows me nothing. It does explain however that common factors are not equal to ...
0
votes
1answer
102 views

Solving Simultaneous Equations $a-ar^2=112$ and $ar-ar^3=84$

I am stuck solving for $a-ar^2=112$ and $ar-ar^3=84$ I got $a=\frac{112}{1-r^2}$ and $a=\frac{84}{r-r^3}$ Then I got a cubic equation. But answer only has 1 value for a & r, so I think there ...
0
votes
1answer
57 views

Have trouble reducing ((a + c)^2 - b^2) / (4a^2c^2 - (a^2 + c^2 - b^2)^2)

I'm trying to reduce the below equation but I'm kind of stuck. This is what I have done so far. $((a + c)^2 - b^2) / (4a^2c^2 - (a^2 + c^2 - b^2)^2)$ --> $(a + c - b) (a + c + b) / (4a^2c^2 - a^4 - ...
-3
votes
2answers
365 views

how to prove the following trig identities?

$\frac{\sin^2 x + \cos^2 x}{\csc x} = \sin x$ $(1 - \tan x)^2 = \sec^2 x - 2 \tan x$ $\tan^2 x - \sin^2 x = \tan^2 x \sin^2 x$ $\frac{\cos^2 x - 1}{\cos x} = -\tan x \sin x$ $\cos x (\tan x + \cot ...
1
vote
1answer
236 views

How to find the sum? Based on logarithm function expansion

The problem: How to find the sum? $$-\sum_{i=1}^{\infty}\frac{(-x)^{i\; \bmod(k-1)}}{i}$$ Details: I tried find this sum using Mathematica ...
2
votes
1answer
106 views

How should i evaluate empty brackets?

I'm working on simple expression evaluation in one competition. Now i'm wondering how should i evaluate empty brackets. I think i should evaluate them as zeros. Is that mathematically right or ...
3
votes
2answers
621 views

Converting multiplying fractions to sum of fractions

I have the next fraction: $$\frac{1}{x^3-1}.$$ I want to convert it to sum of fractions (meaning $1/(a+b)$). So I changed it to: $$\frac{1}{(x-1)(x^2+x+1)}.$$ but now I dont know the next step. ...
-4
votes
1answer
98 views

Inequality question/proof? [closed]

I'm a little confusing in proving this inequality $$\frac{a+b}{|c-b|}<1$$ where $a,b,c$ are positive real numbers, and $a<c$. any help!
1
vote
2answers
251 views

How to write a functional fold in mathematics?

Given is a sequence of natural numbers: $1,2,...,n$. I want to choose two elements $a,b$ of this sequence, calculate $c=ab+a+b$ and write $c$ back to the sequence. After $n-1$ iterations, there is ...
1
vote
1answer
118 views

Given functions $f$ and the composition $h = f \circ g$, how to find $g$?

I have one question just want to be sure that I am correct. Suppose we have two function $f(x)$ and $h(x)$, such that $f(x)$ is linear (i.e., $f(x)=m x+b$) and $h(x)$ is quadratic ...
8
votes
3answers
3k views

Root or zero…which to use when?

This may seem like a very basic question, but: What exactly is the difference between a root of a polynomial, and a zero? Of course I realise that they are technically exactly the same thing, but ...
0
votes
1answer
37 views

Find groups of three

I am trying to come up with an equation that describes groups of three. I have one limitation though, which is that I have to use an ever increasing value of n + 1 for each iteration. Not sure if this ...
3
votes
1answer
910 views

What is the intuition behind the proof of Abel-Ruffini theorem in abstract algebra?

Is there a way to explain this proof in Wikipedia without knowing the abstract algebra too much or deep function experience? In addition, I don't how the abstract algebra work even after I look at an ...
0
votes
2answers
205 views

Guessing a radical expression from a decimal expansion

Is there a function in Maple or Mathematica that takes a truncated decimal expansion, and will try to guess at the value in terms of radicals? Can I use the gfun package in Maple for this? Thanks for ...
1
vote
1answer
215 views

$\sqrt{x+1}+\sqrt{y+1}$ and $\sqrt{x-1}+\sqrt{y-1}$ are non-consecutive integers

We have $$a=\sqrt{x+1}+\sqrt{y+1}$$ $$b=\sqrt{x-1}+\sqrt{y-1}$$ $$x,y>0$$ And we have to find $x$ and $y$ such that $a$ and $b$ are non-consecutive integers. One solution may be 5/4 for both, $x$ ...
1
vote
1answer
158 views

Proof of a formula for the number of distinct roots of a polynomial

I want to proof the following lemma: Given a polynomial $P \in F[X]$ the number of distinct roots is $$d = \deg(P) - \deg(\gcd(P,P')).$$ I see that if $z_1, \dots, z_n$ are the roots and ...
1
vote
2answers
97 views

Sketch the graph of $y = \frac{4x^2 + 1}{x^2 - 1}$

I need help sketching the graph of $y = \frac{4x^2 + 1}{x^2 - 1}$. I see that the domain is all real numbers except $1$ and $-1$ as $x^2 - 1 = (x + 1)(x - 1)$. I can also determine that between ...
12
votes
2answers
961 views

Find all solutions of an exponential equation

Find the product of all the solutions of $\displaystyle\left(\frac{x^2-5x}{6}\right)^{x^2-2}=1$ times the number of solutions. I don't know how to solve an exponential equation, so I've done as ...
2
votes
4answers
138 views

How much percent new content does Grandpa write in the morning?

Grandpa is writing a book. Every morning he starts writing vigorously and fills a lot of pages. But post-lunch he goes through all that he's written that far (right from day one) and deletes one-fifth ...
4
votes
1answer
230 views

Prove for $x$, $y$ and $z$

Well, last week I read a question and I really do not know what to do. I have tried everything I could think, but it does not seem to help me. So, if you can help me, I would be grateful. I'm not ...
2
votes
1answer
101 views

How can I prove this equation: $(1+ \frac{1}{n})^n = 1+ \sum\limits_{k=1}^n{\frac{1}{k!}}1(1-\frac{1}{n})\cdot(1-\frac{2}{n})…(1-\frac{k-1}{n})$ [duplicate]

Possible Duplicate: Proving sequence equality using the binomial theorem $(1+ \frac{1}{n})^n = 1+ ...
2
votes
2answers
102 views

Elementary question on Sums notation

I reading on Sums and I am reading about the difference between using a generalized Sigma notation and the delimited form. Ok, I understand that the generalized form is more expressive. But I ...
0
votes
2answers
3k views

Casio fx-85GT PLUS calculator. How do I access & then use the quadratic equation formula?

Have looked in the user guide - cannot find information on use of the quadratic formula.
17
votes
2answers
14k views

Weird E letter? (sigma) [duplicate]

Possible Duplicate: What does the math notation $\sum$ mean? My school's prescribed book uses the weird letter E character without explaining what it is in the first chapter when it talks ...
3
votes
1answer
202 views

How to find the roots of $f(x)= \ln( \frac{x+1 }{x-2})$?

I can't solve this equation: $$\ln\left(\frac{x+1}{x-2}\right) = 0.$$ I do: $$\begin{align*} \ln \left( \frac{x+1}{x-2} \right)&=0\\ \frac{x+1}{x-2} &= 1 \\ x+1&=x-2 \\ ...
0
votes
1answer
60 views

A question about fractional polynomials (two)

Firstly sorry for this topic's title.. $${P(x)\over x^2}=x-1 \Rightarrow {P^3(x)\over x^2}=?$$
0
votes
1answer
67 views

A question about fractional polynomials

Suppose $$\frac{2(1-2x)}{x^2-x-2} = \frac{A}{x-2} + \frac{B}{x+1}$$ How can I get the value of $A+B$ for my math exam?
2
votes
3answers
233 views

How to convert $\sqrt{\frac{5}{3}}$ to $\frac{\sqrt{15}}{3}$?

Disclosure: This is homework, but not part of the homework. This is just something that I do not understand. $$ x = \sqrt{\frac{5}{3}} $$ $$ x = \frac{\sqrt{15}}{3} $$ Could anyone please ...
2
votes
3answers
134 views

Algebra equations - how to solve?

I have two equations that I want to solve but I can solve the first but not the second, here's an example: $$\begin{align*} 100 &= 120 \times x\\ 0.83 &= 123/ x \end{align*}$$ The ...
1
vote
5answers
274 views

Which is the “fastest” way to compute $\sum \limits_{i=1}^{10} \frac{10i-5}{2^{i+2}} $?

I am looking for the "fastest" paper-pencil approach to compute $$\sum \limits_{i=1}^{10} \frac{10i-5}{2^{i+2}} $$ This is a quantitative aptitude problem and the correct/required answer is ...
1
vote
0answers
72 views

Find $f(k)$ form the given data

Consider a function $f(k)$ defined for positive integers $k=1,2,\cdots \infty$ the function is satisfies the condition $f(1)+f(2)+\cdots = p/(p-1)$, where $0<p<1$, how to determine f(k) from ...
2
votes
3answers
166 views

How to show that $x=\ln 3$ solves $x=\ln(10/3 - e^{-x})$

My homework has tasked me with finding $x$ when $\cosh x=5/3$. I know that the solution is $\ln (3)$, but I can't figure out how to solve it myself. The furthest I can simplify it is the following: ...
1
vote
4answers
95 views

Stuck with logarithm; Find the $x$ value

I am trying to find the value of $x$ ... but I'm absolutely stuck, some hints would be appreciated! $$ \log_3 (6x+2) - 2\log_3 (x)=2 $$ My work so far: $$\begin{align*} ...
4
votes
1answer
95 views

Find x in the following equation

I am trying to find the value of x in the following problem, I have to solve it without logarithm. Problem : $$ \dfrac {27 ^ {(2x+1)} } { 3 ^ {(x+1){5}}} = \dfrac{1}{3} $$ EDIT: My work so far: ...
5
votes
3answers
482 views

Learning basic math?

Not sure if this is appropriate here but I am failing my calculus class and I basically have to take the last year of college over again. College I am trying to transfer to said I don't have the ...
-3
votes
3answers
176 views

I need a proof that this equation has more than one solution [closed]

I need a proof that this equation has more than one solution for $p$ and $q$. $$p^{q-2}= 1024$$, where $q\in N, q>2$
2
votes
5answers
166 views

Can you explain this please $T(n) = (n-1)+(n-2)+…1= \frac{(n-1)n}{2}$ [duplicate]

Possible Duplicate: Proof for formula for sum of sequence 1+2+3+…+n? Can you explain this please $$T(n) = (n-1)+(n-2)+…1= \frac{(n-1)n}{2}$$ I am really bad at maths but need to ...
4
votes
2answers
230 views

How to prove that $x^4+x^3+x^2+3x+3 $ is irreducible over ring $\mathbb{Z}$ of integers?

Which criterion (test) one can use in order to prove that $x^4+x^3+x^2+3x+3 $ is irreducible over ring $\mathbb{Z}$ of integers ? Neither of Eisenstein's criterion and Cohn's criterion cannot be ...
0
votes
1answer
87 views

What term describes the property that of terms can be multiplied or divide in any order?

What term describes the property of terms that can be multiplied or divided in any order? ie, xyz = yxz or x+y+xy+c = c+x+y+yx
2
votes
1answer
151 views

Intersections of 2 circles

Let me ask a similar question to the one I did yesterday. I got answers which said that the following problem had no general solution for x and y. $\sqrt{(n_1-x)^2+(n_2-y)^2}=n_3$ ...
2
votes
1answer
1k views

How many ways a composite number can be resolved into two factors which are prime to each other?

Let N denote the number, and suppose $N=a^p \times b^q \times c^r \cdots$, where $a,b,c,\cdots$, are different prime numbers and $p,q,r,\cdots$ are positive integers.Then it is clear that each term of ...
4
votes
4answers
205 views

How can one prove that $\sqrt[3]{\left ( \frac{a^4+b^4}{a+b} \right )^{a+b}} \geq a^ab^b$, $a,b\in\mathbb{N^{*}}$?

How can one prove that $\sqrt[3]{\left ( \frac{a^4+b^4}{a+b} \right )^{a+b}} \geq a^ab^b$, $a,b\in\mathbb{N^{*}}$?