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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8
votes
3answers
451 views

Prove that $\sum\limits_{k=1}^n \frac{1}{k^2+3k+1}$ is bounded above by $\frac{13}{20}$

I want ask a question about a sum. The exercise is as follows: Prove the following inequality for every $n \geq 1$: $$\sum\limits_{k=1}^n \frac{1}{k^2+3k+1} \leq \frac{13}{20} .$$
1
vote
2answers
275 views

Use division instead of multiplication

Is there any way to calculate x/y without using division? Like 1/10 = 1 * 0.1 . I'm asking because dividing is slower than multiplying in programming programs.
0
votes
2answers
258 views

Counting functions

How many functions are possible from the set $A=\{0,1,2\}$ into the set $B =\{0,1,2,3,4,5,6,7\}$ such that $f(i) \le f(j)$ for $i \lt j$ and $i,j \in A$? I am not sure which counting model ...
0
votes
2answers
99 views

simplify $a=a*b$

Normally in algebra, you can simplify an equation such as $a=a*b$ by dividing both sides by a common coefficient. For example, $\frac {a}{a}=\frac {ab}{a} \rightarrow1=b$. Obviously $b = 1$ is a ...
45
votes
4answers
2k views

A new imaginary number? $x^c = -x$

Being young, I don't have much experience with imaginary numbers outside of the basic usages of $i$. As I was sitting in my high school math class doing logs, I had an idea of something that would ...
1
vote
1answer
280 views

How to solve for the equation $ax \exp(bx)=c$?

How to solve for the equation $ax \exp(bx)=c$? It is known that $x\geq 0$.
2
votes
2answers
3k views

Solving for $x$ when $x$ is the denominator

How do you solve for $x$ when $x$ is in the denominator? E.g. $$10 = \frac{g-1}{x}$$
5
votes
4answers
3k views

Is there a formal name for an equation that has no solution?

I was wondering if there is a formal name for the equations which don't have any solution? For example consider this equation in $m$ : $$ -2(3-m)+15=6m-4(m-20)$$ If we do the algebra we will get ...
0
votes
1answer
90 views

Making $y$ the subject of an equation

How do I make $y$ the subject of the equation $x^3+y^3-3xy=k$?
0
votes
2answers
54 views

Finding degree of expression

In the expansion of (2a − 1)^n, the coefficient of the second term is −192. Find the value of n. How would I work this question out, without brute-forcing every combination? (I got a shocking high ...
2
votes
1answer
671 views

The distance between two points given distances that bees meet in two different time intervals

The original question, actually from a Intro Comp Sci course. Two bees, named romeo and juliet, live in different hives but have met and fallen in love. On a windless spring morning, they ...
5
votes
3answers
298 views

If $a^3+b^3 = c^3+d^3$ and $a^2+b^2 = c^2 + d^2$, then $a + b = c + d$

I came across this problem today. I would be interested to see if anyone knows a proof for it: If $a^3+b^3 = c^3+d^3$ and $a^2+b^2 = c^2 + d^2$, then show that $a + b = c + d$.
0
votes
2answers
112 views

Solve $8^{x+1} = 32 \cdot\sqrt2$ without $\log$

I need help solving the equation $8^{x+1} = 32 \cdot\sqrt2$. The obvious answer is to use log, but that is reserved for the next section. The example given for this section of questions is: $4^x = 8$ ...
1
vote
3answers
140 views

Find x in $4^{\sin^2x}+4^{\cos^2x}=8$

$$4^{\sin^2x}+4^{\cos^2x}=8$$ I solved like this: \begin{align*}4^{\sin^2x}+4^{\cos^2x}=8&\Rightarrow4^{\sin^2x}+4^{1-\sin^2x}=8\\ &\Rightarrow4^{\sin^2x}+\frac{4}{4^{\sin^2x}}=8 ...
0
votes
3answers
47 views

Show that $1/Φ(t) + 1/Φ(-t) = 1$

Let $Φ(t) = 1 + a^t$ Show that $1/Φ(t) + 1/Φ(-t) = 1$ I'm not sure where to start on this one. We've just started exponential functions, so I'm going to assume I just subsitute in $1 + a^t$ for ...
0
votes
1answer
44 views

Need assistance solving the exponential equation

I need to solve the exponential equation $((x + 4)10^x)/(x - 3) = 2x(10^x)$ assuming the fact that $2^x$ is always positive. The example uses the case $x^3-2^x - 3(2^x) = 0$ factors out $2^x$ ...
5
votes
2answers
166 views

Find x: $2^{2x^2}+2^{x^2+2x+2}=2^{4x+5}$

$$2^{2x^2}+2^{x^2+2x+2}=2^{4x+5}$$ I have to find x in this exponential equation. I tried to write it in another way, like this: $2^{2x^2}+2\cdot2^{(x+1)^2}=2\cdot2^{4(x+1)}$ But I don't think ...
3
votes
2answers
76 views

What is the coefficient of $x^{19}$ in the expansion of $ \prod \limits_{n=1}^{20} (x+n^2)$?

How could we find the coefficient of $x^{19}$ in the expansion of $ \prod \limits_{n=1}^{20} (x+n^2)$?
6
votes
1answer
198 views

If $f(x)$ is a polynomial satisfying $ f(x)f(\frac 1x) = f(x)+f(\frac 1x)$ and $f(3)=28$, then how could we find $f(4)$?

If $f(x)$ is a polynomial satisfying $ f(x)f(\frac 1x) = f(x)+f(\frac 1x)$ and $f(3)=28$, then how could we find $f(4)$ ?
1
vote
0answers
86 views

Solving $c(n, n-2) - p(n, 2) = 7 - n$

To help with a friends homework, I was asked how to solve this equation for $n$: $$c(n, n-2) - p(n, 2) = 7 - n$$ Having no further information about what c and p are supposed to be (and they don't ...
3
votes
5answers
507 views

Binomial theorem application

I have a question about the bonomial theorem, and in specifically, a question that I want help on. I have worked out the answer, but by manually expanding each and every alternative. However, I ...
0
votes
2answers
265 views

distributive law in polish notation

On page 18 "Logic as Algebra" Halmos&Givant wrote the distributive law in Polish notation as $$ = \times a + bc + \times ab \times ac $$ I fail to see anything remarkable here, is there a ...
-1
votes
2answers
6k views

Pre Calc, Sinusoidal Function [closed]

On a particular Labor Day, the high tide in southern California occurs at 7:12 am. At that time you measure the water at the end of Santa Monica Pier to be 11 ft deep. At 1:24 pm, it is low tide, ...
0
votes
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
457 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
1k 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
181 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
45k 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
92 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
54 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
344 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
222 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
103 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
591 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
95 views

Inequality question/proof?

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!
0
votes
2answers
227 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
117 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
2k 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
28 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 ...
2
votes
1answer
833 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
199 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
146 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
89 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
880 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 ...