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
votes
1answer
47 views

Reverse an equation with absolute values

I have this line of code in my program: z = (|(x - s_x)| * sin(rot)) + s_z; I have the z variable, I am trying to find out ...
2
votes
1answer
75 views

Minimizing the product $xy$ subject to a polynomial constraint on $x, y$

Given that $$16y(x^2+1)=25x(y^2+1),$$ where $x,y$ are positive integers, find the smallest possible value of $xy$. I wrote my expression as a quadratic in $x$ and calculated it in form of $y$. ...
3
votes
1answer
74 views

Evaluate a rational function of $x,y,z$ given two polynomial equations in $x,y,z$

Let $x, y, z$ be real numbers. Given that $$2x(y^2−1)+2y(x^2−1)=(1+x^2)(1+y^2)$$ and $$4z(1−y^2)+4y(1−z^2)=(1+z^2)(1+y^2)$$ Find the value of the following expression: ...
5
votes
2answers
241 views

Sum of squares in geometric progression

In the geometric progression $b_1, b_2, b_3,\ldots, b_1+b_3+b_5=10$ and $b_2+b_4=5$. Find the sum of the squares of the first five terms. If you solve for the first term and the common ratio, ...
3
votes
3answers
58 views

Prove $\frac{a}c = \frac{a-b}{b-c}$

Suppose $\frac{1}a,\frac{1}b,\frac{1}c$ are three consecutive terms in an arithmetic sequence. Show that: $$\frac{a}c = \frac{a-b}{b-c} $$ and that: $$\frac{2ac}{a+c} = b$$ How would I prove this? ...
0
votes
1answer
16 views

Solve for r. Logarithms

$$ 36000 = 3450 * \frac{1-[1/(1+r)^{12}]}{r} $$ The next step is divide both sides by 3450. Now I'm stuck. Help solve for r.
5
votes
4answers
125 views

Explaining that $1 \cdot 3 \cdot 5 \dotsm (2n+1) = 1 \cdot 3 \cdot 5 \dotsm (2n-1)(2n+1)$

I have a few students that are having trouble understanding that $$1 \cdot 3 \cdot 5 \dotsm (2n+1) = 1 \cdot 3 \cdot 5 \dotsm (2n-1)(2n+1),$$ specifically that $$\frac{1 \cdot 3 \cdot 5 \dotsm ...
1
vote
2answers
49 views

What is the slope of the line that passes through the points (−1,−7) and (−13,−7)? [closed]

I need help ! For finding slope Im getting upset any answers ?
6
votes
2answers
433 views

How long does it take the average voter to vote?

So I was helping my brother with his homework question as follows The voting office can handle $50 \space \text {voters/hour}$ and has 20 voting stations. How long does it take the average voter to ...
0
votes
1answer
45 views

Let $f(x)$ be a function defined for all positive real numbers satisfying the conditions

Let $f(x)$ be a function defined for all positive real numbers satisfying the conditions $f(x) > 0$ for all $x > 0$ and $f(x - y) = \sqrt{f(xy) + 1}$ for all $x > y > 0$. Determine ...
1
vote
3answers
34 views

How do I find an $\varepsilon > 0$ so that $x - \varepsilon > k$ implies $x > k$ ($k$ is a constant)?

Say I have $x - \varepsilon > k$, and we know that $x > k$, and I want to find some positive term for $\varepsilon$ in terms of $x$ so that once I solve $x - \varepsilon > k$ for $x$, I get ...
1
vote
3answers
52 views

Write down values of $a$ and $b$ for which this system of equations has a non unique solution

As a part of a task I got the values of $x$, $y$ and $z$ by solving a system of three equations. The values of $x$, $y$ and $z$ are as follows: $x=\frac {5-2a-3b}{2b}$ $y=\frac {4a+11b-10}{2b}$ ...
2
votes
1answer
41 views

Factoring algebraic expressions of three variables

I want to factor $$bc^2+ab^2+a^2c-b^2c-ac^2-a^2b$$ Using Wolfram, I know it's factored into $$-(a-b)(a-c)(b-c) = (b-a)(a-c)(b-c)$$ However, I don't think I ever got taught how to simplify such ...
7
votes
2answers
180 views

Simple Finite Continued Fraction

I am working on my senior thesis and have encountered, unexpectedly, a finite continued fraction that I would be interested in resolving. I already know the answer (by an informed guess based on where ...
-1
votes
5answers
65 views

I faced a problem proving a statement.

The problem wants me to prove this: $$556^2-445^2=111111$$ $$5556^2-4445^2=11111111$$ $$.$$ $$.$$ $$.$$ ...
0
votes
5answers
92 views

Given $n$, what function returns $0$ for $n < 1$, but $1$ for all else?

I'm looking for a simple operation that returns $0$ if $n$ is less than $1$, but $1$ for anything greater than or equal to $1$. What does the trick?
2
votes
1answer
41 views

Finding polynomial Coefficients

Let $f(x) = x^5 - x^4 + ax^3 + bx^2 + 8x + 4$ The root will make sure, $f(2) = 0$ Which shows: $$2^5 - 2^4 + a2^3 + 4b + 16 + 4 = 0$$ $$16 + 8a + 4b + 20 = 0 \implies 8a + 4b = -36 \implies 2a + ...
3
votes
1answer
122 views

Value of an integral involving the fractional part function

I have difficulties in evaluating the double integral defined in the following. Let $$\left\{ t \right\} = t - \lfloor t \rfloor, $$ $ t> 0$ be the fractional part function, where the ...
0
votes
2answers
84 views

Suppose $f(x)$ is a rational function such that $3 f \left( \frac{1}{x} \right) + \frac{2f(x)}{x} = x^2$ for all $x \neq 0$. Find $f(-2)$. [closed]

Suppose $f(x)$ is a rational function such that $$3 f \left( \frac{1}{x} \right) + \frac{2f(x)}{x} = x^2$$ for all $x \neq 0$. Find $f(-2)$.
5
votes
1answer
100 views

Polynomial maximization

If $x^4+ax^3+3x^2+bx+1 \ge 0$ for all real $x$ where $a,b \in R$. Find the maximum value of $(a^2+b^2)$. I tried setting up ...
1
vote
1answer
34 views

Inverse scaling in which the range [-2.5, 2.5] becomes [3, 1]

I am trying to figure out how to inverse scale a range of -2.5 to 2.5 to the range of 3 to 1. -2.5 should turn to 3 and 2.5 should turn to 1. I have tried this, but the solution is not suitable when I ...
-2
votes
2answers
87 views

Why it is so? Please explain please help me [duplicate]

Why $ \infty - \infty \neq 0 ?$ Please explain
5
votes
3answers
116 views

Find the maximum and minimum of $x^2+2y^2$ if $x^2-xy+2y^2=1$.

Find the maximum and minimum of $x^2+2y^2$ if $x,y\in\mathbb R$ and $$x^2-xy+2y^2=1$$ My attempt: Clearly, since $x^2-x(y)+(2y^2-1)=0$ and $2y^2-y(x)+(x^2-1)=0$, we have that ...
0
votes
2answers
50 views

solve the following equation for x and y

How do I solve the following equation for x and y: $$xy=\frac{(x+y)^2-\frac{35}{x+y}}{3}$$ I tried using the quadratic formula, but can't figure out completely. This is what I had though I could do: ...
5
votes
1answer
73 views

need help proving an interval

I am trying to proof $$\frac {1} {ek} \le \frac {1}{k} (1 - \frac {1}{k} )^{k-1} \le \frac {1}{2k} $$ for k>=2 to prove this I first multiply by k getting $$\frac {1} {e} \le \left(1 - \frac ...
2
votes
2answers
71 views

Find the limit $\lim_{n \rightarrow \infty}(\frac{ \pi }{ 2 }-\tan^{-1}(n^2))\ln(n!)$, is this correct?

$ \color{black}{\begin{align} \lim_{n \rightarrow \infty} (\dfrac{\pi}{2}-\arctan(n^2))\times \ln(n!)) &=\lim_{n \rightarrow \infty} (\dfrac{\pi}{2}-\arctan(n^2))\times\lim_{n \rightarrow \infty} ...
1
vote
3answers
85 views

Is this a legal way to prove an inequality?

I have to prove the following inequality: $(x+y)\sqrt{\frac{x+y}{2}}\geq x\sqrt{y}+y\sqrt{x}$ where $x,y>0$. After squaring both sides I obtain: $(x^2+2xy+y^2)\frac{(x+y)}{2}\geq x^2y+xy^2$ ...
8
votes
4answers
534 views

Simplification of $\sqrt{14} - \sqrt{16 - 4 \sqrt{7}}$

I was trying to simplify $\sqrt{14} - \sqrt{16 - 4 \sqrt{7}}$. Numerical evaluation suggested that the answer is $\sqrt{2}$ and it checked out when I substituted $\sqrt{2}$ in the equation $x= ...
0
votes
2answers
42 views

Solving for $x=0.14$ from $0.32-0.32x=2x$

I want to solve for $x$ on the problem: $$0.32-0.32x=2x$$ I am doing this with the example problem and it shows that the answer is $x=0.14$, but I am not sure how they obtained this answer.
1
vote
1answer
44 views

Find the growth rate for $y=4^{x +1}$

I have a question on my math homework. What's the growth rate for $y=4^{x +1}$? I just need to find the growth rate. I know how to do these kind of problems except for this one. This would be ...
0
votes
0answers
60 views

squeeze theorem - math

I am trying to prove the following: 1/ek <= (1/k)(1-(1/k))^(k-1) <= 1/2k for k>=2 in doing so I tried induction proof, and contradiction and it didn't work, it gets too complicated... Then ...
0
votes
2answers
33 views

Understanding the translation of a graph (horizontally)

I have been having trouble understanding the translation of a graph. I understand the 'rule in the sense that the '$+$' shifts to the left and the '$-$' to the right when dealing with something like ...
0
votes
0answers
34 views

Surface area of hollow shelter

Dexter builds a hollow shelter using blocks. Each block is a cube measuring 1unit by 1unit by 1unit. His shelter is 6 blocks wide 8 blocks long and 4 blocks tall. 1 block thick. What is exterior ...
0
votes
3answers
34 views

Solve this equation.

$$ \dfrac{x^2-1}{x^2+9-6x-1}=\dfrac{x+2}{x-4}-\dfrac5{(x-2)^2} $$ Can you tell me what should I factorize the denominator? I thought to put $$x^2+9-6x-1=x^2-6x+8$$But I suppose they gave it in in ...
40
votes
5answers
3k views

Inequality from Chapter 5 of the book *How to Think Like a Mathematician*

This is from the book How to think like a Mathematician, How can I prove the inequality $$\sqrt[\large 7]{7!} < \sqrt[\large 8]{8!}$$ without complicated calculus? I tried and finally obtained ...
1
vote
1answer
151 views

Proving the Sine Rule with one line.

Working on a general proof of the Law of Sines for ALL Euclidean triangles. Right triangles are easy. Acute triangles are just two proofs of the right triangle. But this is not sufficient for me. I ...
0
votes
2answers
31 views

Compare $(x(4-x))/2$ and $(x-4)/(x-3)$

Here's what I've done: How can I solve the third grade equation? Please try to help me through factorization.
2
votes
2answers
91 views

Show that $4^n + n^4$ is always composite $\forall n > 1$ [duplicate]

I have to show that: $4^n + n^4$ is always composite $\forall n > 1$. I know that composite numbers are integers greater than one but not prime, but I am finding difficult to solve this ...
1
vote
1answer
28 views

Matrix manipulation: Scale by a scalar

I am trying to multiply a Gamma distribution by a Gaussian and the bit that is confusing me is as follows $$ w \exp \big(-\frac{1}{2} (y- \beta x)^T w \Sigma^{-1}(y-\beta x\big) $$ Here $w$ is a ...
1
vote
1answer
88 views

Sum of $p$ terms of an A.P. is $q$, and the sum of $q$ terms is $p$; find the sum of $p+q$ terms.

The sum of $p$ terms of an arithmetic progression is $q$, and the sum of $q$ terms is $p$; find the sum of $p+q$ terms. Answer (as listed at the end of the book): $-(p+q)$
1
vote
1answer
20 views

How to show $\{(x,y)\in \mathbb{R^2}: px+y=1\}$ is unbounded?

I need to show that the set $D=\{(x,y)\in \mathbb{R^2}: px+y=1\}$ is unbounded, where $p>0$ I know this means that I need to show that for all $M>0$, there exits $(x,y)\in D$ such that ...
3
votes
2answers
213 views

Product of Sums: Show that the following is a Polynomial by converting it into standard form. [duplicate]

$$\prod_{k=0}^n (1+x^{2^k})$$ The given expression simplifies to $(1+x)(1 + x^2)...(1 + x^{2^n})$ I am not able to proceed further. How do I express this in Summation form?
4
votes
2answers
42 views

Quadratic residue modulo odd power of $2$

If $x$ and $n$ both are odd positive integers, such that, $$x^2 \equiv -1\mod2^n$$ what can we say about $x$ and $n$ ?
0
votes
2answers
17 views

How does $d=ab-({1\over2}h)^2$ imply: $|d|=({1\over2}h)^2+|ab|$. Hence $|{1\over2}h|<\sqrt{|d|}$, and $|ab|=|d|-({1\over2}h)^2$

Suppose $a,h,b$ are integers where $ab <0$. Suppose we have $d=ab-({1\over2}h)^2$. Considering this, I am having trouble understanding the following implications: $|d|=({1\over2}h)^2+|ab|$. Hence, ...
2
votes
2answers
59 views

If the sum of $n$ terms of an A.P. is $2n+3n^2$, find the $r^{th}$ term.

If the sum of $n$ terms of an Arithmetic Progression is $2n+3n^2$, find the $r^{th}$ term. Note: This question is from the book Higher Algebra by H.S. Hall & S.R. Knight and its answer is ...
0
votes
0answers
42 views

Optimizing a factorization algorithm

In the paper A ONE LINE FACTORING ALGORITHM the following algorithm is presented: ...
1
vote
2answers
68 views

Solve: $-3(6t^3-1)^6 -3t[6(6t^3-1)^5(18t^2)]$

Solve: $-3(6t^3-1)^6 + -3t[6(6t^3-1)^5(18t^2)]$ I don't know how to multiply the two equations and then add them.
0
votes
1answer
35 views

Finding length and width from depth using factors of a cubic equation?

So I have this application question: A pool designer is creating a pool with dimensions of length width and depth that must have specific relationships amongst their scale. Because the design ...
0
votes
3answers
38 views

Solving a function for a variable, confusion

I have the function $f(t) = -4.9t^2+25t+3$, where $f(t)$ is a the height of a grapefruit after $t$ number of seconds. I need to find out how long the grapefruit is in the air, so I know i need to ...
3
votes
3answers
96 views

How can I demonstrate that $x-x^9$ is divisible by 30?

How can I demonstrate that $x-x^9$ is divisible by $30$ whenever $x$ is an integer? I know that $$x-x^9=x(1-x^8)=x(1-x^4)(1+x^4)=x(1-x^2)(1+x^2)(1+x^4)$$ but I don't know how to demonstrate that ...