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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1
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4answers
79 views

The cubic equation $x^3-5x^2+6x-3 = 0$ has solutions $\alpha$, $\beta$ and $\gamma$. [on hold]

The cubic equation $x^3-5x^2+6x-3 = 0$ has solutions $\alpha$, $\beta$ and $\gamma$. Find the value of $$\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}$$
2
votes
2answers
34 views

Knowing that $a,b,c \in ℝ^*_+$ prove that $\frac{a+b}{a+b-c},\frac{b+c}{b+c-a},\frac{c+a}{c+a-b} $ don't belong simultaneously to the interval $(1,2)$

I have to solve the following problem but I don't know how to : Knowing that $a,b,c \in ℝ^*_+$ prove that $\frac{a+b}{a+b-c},\frac{b+c}{b+c-a},\frac{c+a}{c+a-b} $ don't belong simultaneously to the ...
0
votes
1answer
12 views

Simplification and rearrangement of a division of two products

I am quite embarrassed for even considering posting this, so if the solution is childishly simple, my apologies, but I cannot figure it out. Suppose I have this function: $f(a, u, n) = \frac {\prod \...
0
votes
2answers
56 views

Why is $\frac{-125}{-8} = \frac{1}{r^{3}}$ equate to $\frac{8}{125} = r^{3}$?

I am writing exponential functions and I have reached this: $\frac{-125}{-8} = \frac{1}{r^{3}}$ I tried to get an answer but got it wrong, so I opened all the hints and it came to this part of the ...
-1
votes
3answers
46 views

Find the value of k and h [on hold]

$$ \frac{k(4-4)}{(3-1)h} =-1 $$ The numerator will become zero then LHS can never become equal to RHS. What approach will we make to solve these kind of equations and why? Exact Question- Consider ...
0
votes
1answer
44 views

clarify doubts about polynomial

In my math algebra class my teacher says if $$(1+n)^3=A+B(n)+C(n)(n-1)+D(n)(n-1)(n-2)$$ And solve to find A,B,C,D.I know how to solve it. But I won't understand what it really mean and why he says ...
1
vote
1answer
23 views

If $\alpha_i$ are the roots of $x^n + nax−b = 0$ then show that $\prod_{1< i \le n} (\alpha_1 -\alpha_i)=n(\alpha_1^{n-1}+a)$

If $\alpha_i$ are the roots of $x^n + nax−b = 0$ then I would like to show that $$\prod_{1< i \le n} (\alpha_1 -\alpha_i)=n(\alpha_1^{n-1}+a).$$ The only thing I could think is differentiating $x^...
3
votes
1answer
38 views

How to solve the equation $xy = 1, x^{2x-y} = y^{2(x-y)}$

I have the following equation that I don't know how to solve: $$ \begin{cases} xy = 1 \\ x^{2x-y} = y^{2(x-y)} \end{cases} $$ Here's what I've tried (but my mathematical instinct tells me that I didn'...
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0answers
15 views

Sketching polynomial functions [on hold]

On a grid sketch two different polynomial functions which are of degree 5 and belong to the same family. write a general equation which describes the family of functions you have sketched.
1
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1answer
54 views

Equation and roots finding without multiplying parenthesis

Today I'm studying some functions and I have found that equation: $f(x) = (x+1)(x+2)(x-3)$ I solve it by multiplying each parenthesis in order to have, after some addition, an equation like that: $...
0
votes
2answers
59 views

The expression $(x+a)(x+1991) +1$ can be factored as a product $(x+b)(x+c)$ where $b,c$ are integers

I need to solve following problem, but don't know to start with, please provide hints and solutions. Find all integer values of $a$ such that the quadratic expression $(x+a)(x+1991) +1$ can be ...
1
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1answer
40 views

Is there any reason to “anti-simplify” this expression?

I was tutoring a precalculus student, and the question at hand was asking to find the angle between two vectors, given the formula $$\cos \theta = \dfrac {\mathbb{u} \cdot \mathbb{v}}{\|\mathbb{u}\|\...
-6
votes
0answers
24 views

If the system of equation has no solution, and $a$ is a constant, what is the value of $a$? [on hold]

If the system of equation has no solution, and $a$ is a constant, what is the value of $a$? Equations: \begin{align*} ax+y&=-5\\ -\frac{1}{3}x-2y&=-1 \end{align*} Find $a$ and explain.
3
votes
1answer
71 views

What does $\sqrt{2i}$ imply in a question?

While doing a certain problem based on complex numbers I faced this doubt. When $\sqrt{2i}$ is mentioned in a question should I take it's value as $(1+i)$ or both $(1+i)$ and $(-1-i)$ ?I mean should ...
0
votes
2answers
44 views

How to show that $e^{-\frac{1}{x}}<x^n$ within $0<x<\delta$?

I would just like to show that, given a positive integer $n$, it is possible to find a positive real number $\delta$ such that $$e^{-\frac{1}{x}}<x^n,~~~~0<x<\delta$$ For various values of $...
0
votes
1answer
17 views

Pre-Calculus “Function and Graphs” Temperature scales question!

The relationship between the temperature reading F on the Fahrenheit scale and the temperature reading C on the Celsius scale is given by C=5/9(F - 32) a) Find the temperature at which the reading ...
0
votes
1answer
31 views

Maximizing Profit

I have to make a \$25,000 order with 5 SKU'S. The cost I pay are as follows: SKU 1: \$65 SKU 2: \$150 SKU 3: \$218 SKU 4: \$80 SKU 5: \$105 The profit I make on a sale are as follows: SKU 1: \$...
0
votes
3answers
24 views

Cost functionality

I'm thinking that I have encountered a brainfart (I haven't done this type of math in awhile) but I'm stuck on a problem. I'm trying to figure out how much does it cost to produce each speaker system ...
2
votes
3answers
50 views

How does one plug radicals with non-perfect squares and variables into the Pythagorean theorem formula?

I am working on the following integral $$\int\left( 7x^2 - 3 \right)^{\frac 5 2} \, dx$$ I want to use the $\sqrt{u^2 - a^2}$ $u = a\sec\theta$ I know in order to get it into the form that will ...
-1
votes
3answers
44 views

Given function $f(x)$ and $m\in \Bbb{R}$, for what values of $m$ is $f(x)\geq 0$ for all $x\in\Bbb{R}$? [on hold]

Given function $f(x)$ and $m\in \Bbb{R}$, for what values of $m$ is $f(x)\geq 0$ for all $x\in\Bbb{R}$? $$f(x)= \begin{cases} (m-1)x+m & \text{for } x<1\\ x^2+(m-2)x+4-2m & \text{for } x \...
5
votes
1answer
146 views

Are there $a,b \in \mathbb{N}$ that ${(\sum_{k=1}^n k)}^a = \sum_{k=1}^n k^b $ beside $2,3$

We know that: $$\left(\sum_{k=1}^n k\right)^2 = \sum_{k=1}^n k^3 $$ My question is there other examples that satisfies: $$\left(\sum_{k=1}^n k\right)^a = \sum_{k=1}^n k^b $$
0
votes
1answer
33 views

rewriting one expression as other expression

Can anyone explain as how we can rewrite the first expression as second ? I am not able to pick the step done to change from 1 to 2. $f(N) = 2^{(N+1)/2 + 1 }- 2$, for odd $N$ becomes $f(N) = 2^{(N+...
0
votes
1answer
32 views

If $(\sqrt{x^2-5x+6} + \sqrt{x^2-5x+4})^{x/2} + (\sqrt{x^2-5x+6} - \sqrt{x^2-5x+4})^{x/2}=2^{\frac{x+4}{4}}$, find $x$.

The main question is : If $(\sqrt{x^2-5x+6} + \sqrt{x^2-5x+4})^{x/2} + (\sqrt{x^2-5x+6} - \sqrt{x^2-5x+4})^{x/2}=2^{\frac{x+4}{4}}$, find $x$. My method : I first began by substituting $x^2-5x+5$ as ...
-2
votes
1answer
29 views

How do you calculate what $k$ is? [on hold]

How do you calculate what $k$ is? $$\frac{d-1}{k} = \log \left(\frac{21}{100} d\right)$$ And how do I get the integer value?
0
votes
3answers
46 views

Series question involving a cubic polynomial

The question asks: Consider the polynomial $\displaystyle{\,\mathrm{f}\left(X\right) = X^{3} -6X^{2} + mX - 6}$, where $m$ is a real parameter. a. Show that: $\displaystyle{{1 \over x_{1}x_{2}} ...
4
votes
6answers
276 views

How to Find a rational number between two irratonal number? [on hold]

Find the rational number between $\sqrt 2$ and $\sqrt3$. I try to solve by using some methods in my book but can not understand steeps.
4
votes
5answers
54 views

Find two different rules for one sequence $2, 4, 8, \ldots$

Question: The first three terms of a sequence are given. $2, 4, 8...$ Write two different rules for continuing the sequence. Give the next two terms for each rule. Answer: I have found one rule ...
1
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3answers
34 views

Problem related to series of binomial coefficients

Problem related to series of binomial coefficients in which each term is a product of two binomial coefficients. In this question: Prove that $$\binom{n}0^2+\binom{n}1^2+\ldots+\binom{n}n^2=\...
0
votes
1answer
17 views

Finding the equation of a hyperbola given focus and asymptote

Find the equation of a hyperbola with the focus at $(5,1)$ and asymptote at $y=1\pm 2x$ I know that the hyperbola would be a vertical one, since the asymptote is at $y$. The asymptote's formula is $$...
0
votes
1answer
15 views

Prove that: ${^{n}\mathrm{C}_{k}} = {^{n-1}\mathrm{C}_{k-1}}+{^{n-1}\mathrm{C}_{k}}$ [duplicate]

Question asks to prove: ${^{n}\mathrm{C}_{k}} = {^{n-1}\mathrm{C}_{k-1}}+{^{n-1}\mathrm{C}_{k}}$ My Steps: $$\begin{align*}\frac{(n-1)!}{(n-k-2)!(k-1)!} + \frac{(n-1)!}{(n-k-1)!(k)!} & = \...
1
vote
2answers
43 views

Rationalizing $\frac{\left(\frac{1}{\sqrt{x}}\right)-1}{x-1}$

I'm working through some rationalization problems and came across this problem: $\frac{\left(\frac{1}{\sqrt{x}}\right)-1}{x-1}$ The answer is given as: $\frac{-1}{\sqrt{x}+x}$ I can't for the life ...
2
votes
1answer
23 views

Simplification of probability expression

Let $p_1$, $p_2$, and $p_3$ be probabilities such that $p_1 + p_2 + p_3 = 1$, and let $c_1$, $c_2$, and $c_3$ be arbitrary constants. Can the following expression be written in terms of $p_1$ and ...
1
vote
1answer
36 views

$f(x) = -2x^2 + 10x - 8$

For $f(x) = -2x^2 + 10x - 8$, labeling any intercepts and the vertex and showing the axis of symmetry. I came up with $(0, -8)$, $(4, 0)$, $(1, 0)$, and the vertex $(2.5, 4.5)$. The axis of symmetry ...
-3
votes
0answers
26 views

ratio of derivatives [on hold]

({deriv(y, x)}^7derivN(x, y, 3)+{deriv(y, x)}^3derivN(y, x, 3))/({deriv(y, x)}^2{derivN(y, x, 2)}^2) When I looked at this problem I thought the problem is incorrect as we have not been given y as a ...
0
votes
1answer
26 views

What is the greatest remainder if you divide a 2-digit number by its digit sum

I just found this problem and tried to solve it. I wrote $x=90a+b$ and tried to maximize the function $f(a,b)=\frac{9a+b}{a+b}$ but did not come to any solution. Then I considered $10a+b = x\pmod{a+...
2
votes
2answers
41 views

Algorithm for multiplying infinite decimals?

What is the (best) algorithm for multiplying two real numbers based on their decimal expansions? Obviously the algorithm can't be completed but I mean an algorithm that will successively approximate ...
1
vote
2answers
32 views

What is the value of xyz?

If $a,b$ and $c$ are not equal to $0$ and $1$ and if $a^x=b,b^y=c,c^z=a$,then $xyz=?$ We have tried to solve by equation,but it can't produce the desired result.
1
vote
2answers
24 views

Pairs $x_i+x_j$ positive for total positive sum

Let $x_1,\dots,x_n\in\mathbb{R}$ be such that $x_1+\dots+x_n>0$. At least how many sums $x_i+x_j$ ($i<j$) must be positive? It is possible that $x_1=n$ and $x_2=\dots=x_n=-1$, in which case $n-...
1
vote
3answers
38 views

Calculating integer solutions of a logarithmic equation

The question asks: Calculate the integer solutions of the equation $\log_2(x+2)+\frac12\log_2(x-5)^2=3$ To me, this is trivial if solved in the following way: $(x+2)+(x-5)=2^3$ $2x-3=8$ Answer: $x=\...
1
vote
1answer
33 views

Simplifying $\frac{^n\mathrm{C}_k}{^n\mathrm{C}_{k-1}}$

Question asks to simplify: $$\frac{^n\mathrm{C}_k}{^n\mathrm{C}_{k-1}}$$ I have a few steps but not sure if its correct. $$\begin{align*}\frac{(n)!}{(n-k)!(k)!} \bigg/ \frac{(n)!}{(n-k-1)!(k-1)!}...
0
votes
2answers
49 views

What is the coefficient of $x^{2m}$ in $(1 + 4 x - 2 x^2 + 4 x^3 + x^4)^m$?

For each positive integer $m$, write $(1 + 4 x - 2 x^2 + 4 x^3 + x^4)^m = \sum_{j = 0}^{4m} b_j^{(m)} x^j$. What is $b_{2m}^{(m)}$ in terms of $m$?
-3
votes
2answers
60 views

simplify factorials: $\frac{(k-1)!}{(k+2)!}$ [duplicate]

Question: simplify $$\frac{(k-1)!}{(k+2)!}$$ What I did was: $$\frac{(k - 1)!k!}{(k + 2)! \cdot (k + 1)!}$$ This I did following the rule $n! = n \times (n - 1)!$. can this be simplified ...
0
votes
2answers
64 views

Simplifying factorials: $\frac{(n-1)!}{(n-2)!}$

Question: simplify $$\frac{(n-1)!}{(n-2)!}$$ What I did was: $$\frac{(n - 1)!}{(n - 2)! \times (n - 3)!}$$ This I did following the rule $n! = n \times (n - 1)!$. But my answer just doesn't look ...
0
votes
1answer
14 views

Divide items with integer ID-s into N equal groups, based on ID-s

I have unknown number of items, each having ID (consecutive integer numbers), ie. 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15... I want to split above items into as ...
0
votes
1answer
31 views

Finding a rule for the outcome of $\sqrt[4]{r}(a+\sqrt{r})$

It is well known that $$\sqrt{4+3\sqrt{2}}=\sqrt[4]{2}(1+\sqrt{2})\tag{1}$$, and similarly, $$\sqrt{10+6\sqrt{5}}=\sqrt[4]{5}(1+\sqrt{5})\tag{2}$$$$\sqrt{6+4\sqrt{3}}=\sqrt[4]{3}(1+\sqrt{3})\tag{3}$$$$...
0
votes
1answer
25 views

Hoses in the Road [on hold]

Perhaps you have seen the hoses in the road measuring traffic flow and vehicle speed. A car traveling over them will first hit the back hose and then the front hose. A computer measures the time ...
0
votes
2answers
25 views

Finding the domain and range of a difficult piecewise composite function

I recently inquired about finding a formula for a composition of two piecewise functions, but I have been thoroughly confused by a slightly different example. In this case, I have a question about ...
1
vote
1answer
24 views

Need help with inductive proof of Binomial Theorem

I'm new to math and trying to learn about the Binomial Theorem, by following this tutorial. I got stuck trying to read the Induction Proof. They give an example of using the Sum notation: $$ (x + y)^...
0
votes
2answers
31 views

finding out total digits in a large number

Is there any easy way to find out how many digits does the number $12^{400}$ have or such types of problems like how many digits the number $x^y$ have? ($x$ and $y$ are variables)