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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5answers
71 views

Proof of reciprocal logarithm

I need to prove this logarithm. $$\log_p\Big(\frac{1}{x}\Big) = -\log_p(x)$$ The first step would be $\ln(1/x)/\ln_p$ I need help as to what the next step would be.
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0answers
21 views

Understanding Ferrari's Solution

I'm trying to understand how Ferrari's Solution works. Thanks to this post I understand that we are solving this for $y$ to find the perfect square: $$y^3 + \frac{5\alpha y^2}{2} + (2\alpha^2 - ...
1
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1answer
27 views

Magnitude of one quadratic greater than another

Let $P,Q$ be quadratic polynomials with discriminants $p,q$ such that $|P(x)|\geq|Q(x)|$ for all real numbers $x$. Prove that $|p|\geq |q|$. Suppose that $P(x)=ax^2+bx+c$ and $Q(x)=dx^2+ex+f$. Then ...
4
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1answer
62 views

Jumping in two-dimensional space?

I came up with this problem and have no idea how to approach it: assume that a bug starts at $(0,0)$ and at every second, it jumps in one of the four directions. At second $i$, it jumps a distance of ...
3
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2answers
65 views

How many complex roots does this polynomial $x^3-x^2-x-\frac{5}{27}=0$ have?

The fundamental theorem of algebra states that any polynomial to the $n^{th}$ will have $n$ roots (real and complex). But I know that complex roots only come in pairs because they are conjugates of ...
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0answers
28 views

Determine maximal addend in Newton Binomial Expansion.

Determine the maximal addend in Newton Binomial Expansion of the expression $$\left ( 2n+\frac{1}{2n} \right )^{4n+1},\quad \left ( \forall n \in \mathbb{N} \setminus \left \{ 1 \right \} \right )$$ ...
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2answers
62 views

How to solve the equation $(25{ x }^{ 2 }-1)(10x+1)(2x+1)=11$? [closed]

How to solve this equation? $$(25{ x }^{ 2 }-1)(10x+1)(2x+1)=11$$
1
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5answers
103 views

Factor Theorem and Long Division

For instance Given $f(x) = 2x^3-7x^2+2x+3$, and $(x-3)$ is one of the factors, how do I obtain $2x^2-x-1$ as the quotient without using the long division. Is there any other method apart from long ...
1
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5answers
49 views

Square root math and squaring

I'm having some trouble making sense of this. $\sqrt{\dfrac 12 \operatorname{in.}^2} = \dfrac{1}{\sqrt 2} \operatorname{in.} = \dfrac{1}{\sqrt 2} \operatorname{in.} \times \dfrac{\sqrt 2}{\sqrt 2} = ...
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1answer
53 views

Definition of Cross Product

I am teaching Pre-Cal at a community college. In all Pre-Cal textbooks, we have this definition of cross product of two vectors: If $\textbf{u} = \langle a_1, a_2, a_3 \rangle$ and $\textbf{v} = ...
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1answer
35 views

Solving a Binomial Sextic

Say I have a sextic equation, but I'm able to get it into the form: $$ax^6 + dx^3 + g = 0$$ I know that I can do a simple substitution like $y = x^3$ to get an equation that I can solve with the ...
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2answers
53 views

If the sum of dates within a week is $72$, what date is Monday?

There is a man who spends exactly same no. of dollars as the date (e.g. on $12$ of any month he will spend 12 dollars). In a week he spends 72 dollars. What is the amount spend by him on Monday? I ...
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0answers
17 views

Algorithm for complex roots of high degrees.

Is there an algorithm to find complex roots of equations of high degrees? Let's suppose I'm given an even function of degree greater than 6 that does not have real roots, how am I supposed to find its ...
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5answers
58 views

What is the value of $ x(\log x)$ when $x=0$ and $x\not \to 0$?

I know that $ x(\log x)\to 0$ when $x \to 0$, but some people say that since $\log x$ is not defined when $x=0$, so the value of $ x(\log x)$ can also not be found when $x=0$. But isn't it true ...
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2answers
32 views

Positive equilibria for a system of eqautions

I have the following system of equations \begin{align} \frac{dx}{d \tau} &= x \left(1-x-\frac{y}{x+b} \right) \\ \frac{dy}{d \tau} &= cy \left(-1+a\frac{x}{x+b} \right) \end{align} I am ...
4
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2answers
69 views

How to rationalize denominator?

Suppose $c$ is not a complete square integer, ${a_0},{a_1} \in \mathbb{Q}$, we have $$ \frac{1}{{{a_0} + {a_1}\sqrt c }} = \frac{{{a_0} - {a_1}\sqrt c }}{{a_0^2 - a_1^2c}}. $$ We need to show ...
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4answers
35 views

Simplifying Product of Logarithms

I'm trying to simplify $$\log_{10}11 \cdot \log_{11}12 \cdot \log_{12}13 \cdot \ldots \cdot \log_{998}999 \cdot \log_{999}1000\ .$$ I have absolutely no clue where to start in simplifying it. I ...
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4answers
56 views

Simplify $2 \sqrt[3]{50x^2 z^5} × 3 \sqrt[3]{15 y^3 z}$

Image of problem As you can see the answer is $30 y z^2 \sqrt[3]{6x^2}$ . I understand mostly everything in the problem, but one thing that I am having confusion on is where the "30" came from in the ...
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2answers
15 views

Comparing limits of sequences, where their ratio diverges to $\infty$.

Say: $$\lim_{n \to \infty} \frac{a_n}{b_n}=\infty$$ I know intuitively that for a large enough $n$: $$a_n > b_n$$ So I would think that it be true to say that if: $$\lim_{n \to \infty} b_n= ...
1
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2answers
59 views

Factorising $t ^6 − 10t ^4 + 31t ^2 − 30$

Does anyone know how I can factorise $t ^6 − 10t ^4 + 31t ^2 − 30$? I can see the answer using WolframAlpha but I want to know how to do it by hand without guessing roots.
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3answers
213 views

Best books for relearning precalculus maths the right way

I am looking for advice on the best texts to remind myself how to do math. I am 35 and an attorney (patent) but was once very good at math (best in my high school, 800 SAT) but then stopped completely ...
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2answers
52 views

Finding x when the exponents and coefficients are different

How can I find $x$ when the exponents and coefficients are different? $$3x \cdot e^{7x+2} = 15$$ Thanks
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3answers
329 views

Solving Cubic when There are Known to be 3 Real Roots

When solving for roots to a cubic equation, the sign of the $\Delta$ tells us when there will be 3 distinct real roots (as long as the first terms coefficient, $a$, is non-zero.) Namely when $\Delta$ ...
0
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2answers
34 views

Can we deduce anything given the equation of a curve and the fact that it has symmetry with $y=x$?

Question: The line $y=x$ is a line of symmetry to the curve with equation $$y=\frac{px+q}{rx+s}$$ where $p,q,r,s \neq 0$. Which of the following must be true? $p+s=0$ $p+q=0$ ...
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3answers
52 views

Express $\frac{9x}{(2x+1)^2(1-x)}$ as a sum of partial fractions with constant numerators. Answer doesn't match with solution provided in book.

Express $\frac{9x}{(2x+1)^2(1-x)}$ as a sum of partial fractions with constant numerators. Answer doesn't match with solution provided in book. My method: ...
4
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4answers
124 views

What is the remainder when $1^6 + 2^6 + 3^6 + … + 99^6 + 100^6$ is divided by 5?

What is the remainder when $1^6 + 2^6 + 3^6 + ... + 99^6 + 100^6$ is divided by 5? I think that the only way to solve this would be to applying to the proposition that “the sum/product of congruence ...
1
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1answer
78 views

Finding All 3 Roots of a Cubic

I'm trying to find all real roots of a cubic. I wanted to use Cardando's Method but I'm not sure I'm correctly understanding how to obtain all 3 roots given the depressed cubic: $$t^3 + pt + q = 0$$ ...
0
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3answers
282 views

Which of the following are recursive formulas for the nth term of the following geometric sequence?

$$\frac{3}{4}, 1, \frac{4}{3}, \frac{16}{9}....$$ Please could someone help me, I've been stuck on this question all night. P.S. I'm $13$.
2
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2answers
110 views

Why aren't all differential quantities equal?

Since differential quantities are defined as any variable /function tending to zero ($\lim_{x\to0} x= dx$). This is basically the smallest value that we can imagine. Doesn't this mean that there is ...
2
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2answers
32 views

Confused With Simplifying Exponents

If you had $x^{2/4}$, would that simplify to $x^{1/2}$? If you were to simplify $x^{2/4}$ to $x^{1/2}$, x cannot be negative for a real solution... But if you left it as $x^{2/4}$, x would be able ...
0
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1answer
20 views

howto find where a slant aymptote crosses a function

i have the function $f(x) = \frac {(x+1)(x-1)^2}{x^2}$ and i want to sketch it so i found the following: Vertical Asymptote: is x=0 X-intercepts: x=1, x=-1, x=1 Slant Asymptote: $y = x-1 $ then i ...
2
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1answer
29 views

Finding transcendental roots to an algebraic equation

So for equations with rational roots, there's a theorem that lists all the possible roots (Rational Root Theorem). If an equation has imaginary or irrational roots, their respective theorems say ...
2
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2answers
46 views

Is it possible to find solution of this system of equations?

Following is augmented matrix which has been reduced to row echelon form by using row operations. So when I convert it to system of equations I would get 3 ...
1
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1answer
78 views

Can a function ever cross a vertical asymptote?

I have the function $f(x) = \frac {x^2-1}{x^2-4}$ that I need to graph. This is what I found: Vertical asymptotes: $x=2$ and $x=-2$ Horizontal asymptote: $y = 1$ $x$-intercepts: $x = 1$ and $x = ...
0
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1answer
79 views

Solving line intercept equation for exponential decay using two points?

I have two points (x1,y1 and x2,y2) that represent points in an exponential decay curve (discounted cash flows): Exponential Decay using varying Discount Rates The limits of my mathematics is using ...
0
votes
1answer
21 views

How much time need for finish the work

Two employees can perform a job for $\frac{20}3$ hours. How much time is needed to see his worker to perform work , if for him needs $3$ hours less than the second worker would own work. My attempt ...
1
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3answers
54 views

computing the product $\prod_{n=1}^{2016} \frac{2^{2^{n-1}}+1}{2^{2^{n-1}}}$

how can i calculate the product: $\prod_{n=1}^{2016} \frac{2^{2^{n-1}}+1}{2^{2^{n-1}}}$? I can see that in the denominator it's a geometric series, but in the numerator i can't see how to simplify. ...
6
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2answers
73 views

Solve $x-\lfloor x\rfloor= \frac{2}{\frac{1}{x} + \frac{1}{\lfloor x\rfloor}}$

Could anyone advise me how to solve the following problem: Find all $x \in \mathbb{R}$ such that $x-\lfloor x\rfloor= \dfrac{2}{\dfrac{1}{x} + \dfrac{1}{\lfloor x\rfloor}},$ where $\lfloor *\rfloor$ ...
3
votes
1answer
75 views

Sum up a function series $f(1/9)+f(2/9)+\dots+f(26/9)$ for $f(x)=\frac{9^{x}}{9^{x}+27}$

Given $f(x)=\dfrac{9^{x}}{9^{x}+27}$. Find: $$S=f\left(\frac{1}{9}\right)+f\left(\frac{2}{9}\right)+\dotsb+ f\left(\frac{26}{9}\right)$$ Teacher did not allow us to use calculator...Use sigma ...
0
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1answer
38 views

Getting Equation to Find a Perfect Square

I'm trying to work through the explanation of Ferrari's Solution of the Quartic. The arbitrary variable: $y$ is introduced to the depressed quartic in the link's step 3, yielding a right side of: ...
3
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2answers
31 views

An Algebraic Proof that $|y^3 - x^3| \ge |(y - x)|^3/4 $

I can prove this using calculus, but not by simple algebra: can anyone help ? Calculus Proof: Fix the separation of $x$ and $y$ so that $y = x + d$ with $d>0$ ($ \implies y > x \implies y^3 ...
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1answer
50 views

A student had to solve $26$ problems.how many problems successfully solved and how many wrong? [closed]

A student had to solve $26$ problems.His father promised that he will give him $800$ euro for every problem that we would solve correctly,but he would abstract $500$ euro for every problem that would ...
1
vote
1answer
78 views

Evaluating a double sigma

Evaluate $$\sum_{m=0}^{\infty} \sum_{n=0}^{\infty}\frac{m!n!}{(m+n+2)!}$$ How do I start with the problem? Infinite sum of factorials?
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0answers
31 views

Finding the equation of a compound discount curve using two points.

I have a problem that goes beyond what I am capable of resolving. Basically I have two Net Present Values at different discount rates for a series of UNEVEN cash flows. As a reminder this is the NPV ...
1
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1answer
19 views

Simultaneous equation/expression re-arrangement

I'm having a little difficulty (being a physicist). I have the following two equations (the constants have been changed, but it is identical to the original expression which relates to the virial ...
2
votes
3answers
71 views

How can this sum be maximized?

Suppose that $a_1, a_2, a_3, a_4, a_5, a_6, a_7$ are distinct integers from $1$ to $7$. What is, then, the maximum value of the sum $$|a_1-a_2|+|a_2-a_3|+|a_3-a_4|+|a_4-a_5|+|a_5-a_6|+|a_6-a_7|+a_7$$? ...
1
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1answer
67 views

Finding an equation in $x$ and $y$ with no square roots

I have the equation $$ x^2 + y^2 = 2\left(\sqrt{x^2 + y^2}\,\right) + 2y + 3x $$ and I want to solve it in terms of $y$ with no square roots. I'm having trouble with proceeding from here. My initial ...
2
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2answers
67 views

The roots of $x^2-2x+3=0$ are $\alpha$ and $\beta$. Find the equation whose roots are: $\alpha+2$, $\beta+2$. Not sure of answer in book.

The roots of $x^2-2x+3=0$ are $\alpha$ and $\beta$. Find the equation whose roots are: $\alpha+2$, $\beta+2$. Not sure of answer in book. My working: $\alpha+\beta=2, \alpha\beta=3$ ...
0
votes
1answer
24 views

Simple formula for cumulatively adding multiples of five

I'm a software developer, trying to write a simple formula for the following: A person is tested and given a score (out of 100, but irrelevant). The number of criteria they fail adjusts their score ...
0
votes
1answer
51 views

Taking the square root of this expression.

This expression is written under a radical and I need to take it out. How to take the square root? Simply calculate this: $$\sqrt{(a+b+c)(a+b-c)(a-b+c)(-a+b+c)}\ \ \ \ \ \ \ \ \ (0<a\le b\le ...