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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2
votes
6answers
183 views

Proof that $n^3-n$ is a multiple of $3$. [duplicate]

I'm struggling with this problem of proof by induction: For any natural number $n$, prove that $n^3-n$ is a multiple of $3$. I assumed that $k^3-k=3r$ I want to ...
1
vote
2answers
51 views

Can this be solved for A?

I am trying to solve an equation and it has been a looong time since I've been in math class. $$((c+r+a) - (c+.9r)) / (c+r+a) = 7\%$$ Can this be solved for $a$? Thank you in advance Gman
1
vote
3answers
215 views

What does “$f(x,y)$ is strictly increasing in each argument” imply?

Say we have a function $f(x,y)$. below are what we know about $f(x,y)$ strictly increasing in each argument. $x$ and $y$ are natural numbers only, i.e., $0, 1, 2, ...$ Now we have a fixed number ...
3
votes
2answers
91 views

Finding the values of $a$ and $b$ [duplicate]

We are given two equations: $$ \left\{ \begin{array}{c} \sqrt{a}+b=11 \\ a+\sqrt{b}=7 \\ \end{array} \right. $$ How to find the value of $a$ and $b$ by solving the equation? All I could do was use ...
0
votes
1answer
46 views

inequality of sums

I think about the following inequality: $$ \sum_{i=1}^n a_ib_i \le \left(\sum_{i=1}^n a_i \right)\left(\sum_{i=1}^n b_i \right) $$ Is the inequality true for all $a_i$ and $b_i$, or just correct ...
4
votes
3answers
118 views

Find $x$ for $\left(\frac1{1\times101} + \frac1{2\times102} + \dots +\frac1{10\times110}\right)x = \frac1{1\times11} + \frac1{2\times12}…$

$$\left(\frac1{1\times101} + \frac1{2\times102} + \dots +\frac1{10\times110}\right)x = \frac1{1\times11} + \frac1{2\times12} + \dots +\frac1{100\times110}$$ Find x My younger sister in grade 5 had ...
4
votes
2answers
273 views

What is a good example to show high school students why a proof for induction is a reasonable kind of proof?

I teach average-level high school students who have not had much beyond Algebra 1. I want to show them why induction makes sense. I want the sort of problem where it is intuitive that a statement is ...
1
vote
2answers
31 views

$a\in\mathbb C$ such that $a^n=1,a^m\neq 1\,(m=1,2,\ldots,n-1) $

I would appreciate if somebody could help me with the following problem Q: Suppose $a\in\mathbb C $ is a complex number such that: $$a^n=1,a^m\neq 1\,(m=1,2,\ldots,n-1) $$ find the value of: ...
1
vote
1answer
42 views

Real $(x,y,z)$ in fractional part and greatest integer equation

Calculation of Real $(x,y,z)$ in $x[x]+z\{z\}-y\{y\} = 0.16$ $y[y]+x\{x\}-z\{z\} = 0.25$ $z[z]+y\{y\}-x\{x\} = 0.49$ where $[x] = $ Greatest Integer of $x$ and $\{x\} = $ fractional part of x ...
1
vote
2answers
83 views

If $3^n+81$ is a perfect square, then positive integer value $n$ is

If $3^n+81$ is a perfect square, Then calculation of a positive integer value of $n$. $\bf{My\; Try}::$ When $n≤4,$ then easy to know that $3^n+81$ is not a perfect square. Now let ...
2
votes
1answer
131 views

Finding aggregate score from incomplete data

$$ \begin{align} n & = \text{number of reviews}\\ x & = \text{review score}\\ \bar{x} & = \text{aggregate score} \end{align} $$ I have a specified number of reviews for a product, I have ...
4
votes
4answers
69 views

Proving $\frac{(n+1)^4}{4}+(n+1)^3\le\frac{(n+2)^4}{4}$ for all $n \ge 1$.

$$\frac{(n+1)^4}{4}+(n+1)^3\le\frac{(n+2)^4}{4}$$ For all $n\ge 1$. I thought that I could get rid of the denominators like this: $$(n+1)^4+4(n+1)^3\le(n+2)^4$$ Then, maybe, take $(n+1)^3$ as ...
3
votes
2answers
1k views

Is it possible to express $\sin \frac{\pi}{9}$ in terms of radicals?

So, yes, this is a math homework question. I've done some research on it and I know that the actual value for $\sin \frac{\pi}{9}$ cannot be expressed without using imaginary numbers. ...
0
votes
1answer
70 views

How do you change an eqaution to y=ax^2+bx+c (standard form) Algebra

I have to take this equation: $y=(x-3)^2-2$ and turn it into $y=ax^2+bx+c$ (standard form). Can anyone help me step by step how to change this? Edit: Math tex-style
3
votes
6answers
1k views

Is it possible to have a quadratic equation having only one complex root? If so, what would a picture of it look like?

I'm also wondering the same questions about a quadratic function with two real roots, and a quadratic function with two pure imaginary roots. Is it possible? And if it is possible, what would it look ...
1
vote
4answers
49 views

Inequalities In Algebra

So the problems ask to find Find all values of $x$ for which $$\frac{x}{x-4}<\frac{x-5}{x+1}.$$ So the solution requires moving both fractions to one side, finding a common denominator, ...
0
votes
3answers
179 views

Prove that a circle has an infinite number of tangents

It seems obvious that a circle is comprised of the set of all points that are equidistant from one point, and that each point on the circumference of the circle represents a tangent. This seems to ...
2
votes
2answers
62 views

Fast way to prove that $(a^2+b^2+c^2-ab-ac-bc)^2=(a-b)^2\times (a-c)^2 + (b-c)^2\times(b-a)^2 + (c-b)^2\times(c-a)^2$

What is the most simplest way to prove that $$(a^2+b^2+c^2-ab-ac-bc)^2=(a-b)^2\times (a-c)^2 + (b-c)^2\times(b-a)^2 + (c-b)^2\times(c-a)^2$$ Please!! Thanxx
0
votes
3answers
50 views

Sequence difference equatiom

For $n \ge 2$ the terms in the sequence $a = \{1, 6, 17, 45, 118, 309, \ldots\}$ are related by the difference equation $$a_{n+2} = \boxed{\phantom{XX}} \, a_{n+1} + \boxed{\phantom{XX}} \, a_n $$ ...
-1
votes
1answer
700 views

How to round off timestamp in milliseconds to nearest seconds?

How to round off the current timestamp in milliseconds to seconds? If this is the current timestamp in milliseconds I have - 1384393612958 The if I am rounding ...
1
vote
3answers
63 views

Find $x^4+y^4$ and $x^3+y^3$ if $x+y=2$ and $x^2+y^2=8$

Find $x^4+y^4$ if $x+y=2$ and $x^2+y^2=8$ So i started the problem by nothing that $x^2+y^2=(x+y)^2 - 2xy$ but that doesn't help! I also seen that $x+y=2^1$ and $x^2+y^2=2^3$ so maybe $x^3+y^3=2^5$ ...
0
votes
2answers
136 views

Simple Log Question

Fairly new to logs, stuck here: The equation $y=4^{3x}$ can be written in terms of $x$: $$y=4^{3x}$$ $$\log(y)=\log(4^{3x})$$ $$0=3x\log(4)-\log(y)$$ $$0=3x\log\bigg(\dfrac{4}{y}\bigg)$$ At this ...
1
vote
1answer
170 views

Finding Real/Imaginary part of a square root

I'm not sure if this is obvious or not; say I have the function $$f(z)=\left(\frac{z^2}{4}-c^2\right)^{\frac{1}{2}}$$ where $c$ is constant. Is it possible to expand this into its real and imaginary ...
0
votes
2answers
49 views

Finding the asymptotes of a hyperbola

Given the hyperbola $x^2/16$ - $y^2/9$ $ = 1$ what would the equation of the asymptotes be?
2
votes
3answers
100 views

Solve $x^2+(-7-4i)x+9+15i=0.$

Solve $$x^2+(-7-4i)x+9+15i=0.$$ Using the quadratic formula, I get $$\frac12 (7+4i \pm \sqrt{-4i})$$ but that's not correct. How do you solve this? I get no help from looking at wolfram alpha.
-1
votes
2answers
86 views

The inverse function of a logarithm equation

I've tried many things with this question, and just can't seem to get it quite right, can someone please show me how to answer this question? Thank you in advance. $$g(x) = \ln(5x+25) \qquad g^{-1}(x) ...
18
votes
8answers
1k views

Find $x$ such that $\sqrt{x+\sqrt{x+7}}\in \mathbb{N}$

Find $x$ such that $$\sqrt{x+\sqrt{x+7}}\in \mathbb{N}$$ I tried many ways: $$\sqrt{x+\sqrt{x+7}}=n$$ $$\sqrt{x+\sqrt{x+7}}^2=n^2$$ $$x+\sqrt{x+7}=n^2$$ then solve for $x$ but didn't do with ...
0
votes
1answer
131 views

Word Problem Using elimination and/or substitution

A doctors prescription calls for the creation of pills that contain 12 units of vitamin B and 12 units of vitamin E. Your pharmacy stocks two powders that can be used to make these pills: one contains ...
1
vote
1answer
134 views

For what values of m does the equation 35530x + 355y = m have integer solutions?

For what values of $m$ does the equation $35530x + 355y = m$ have integer solutions? (only find the $m$'s for which solutions exist)
2
votes
1answer
56 views

moving particle gone crazy

Let's say that a point particle moves along a line? We can come with a function describing the speed of the particle. It is not hard to imagine what would the motion of the particle look like if our ...
5
votes
4answers
8k views

What's the algebraic property where you can flip the fractions in an equation?

Earlier in algebra, we spent over 20 minutes trying to figure out $$ \frac{1}{R_1} + \frac{1}{R_2} = \frac{1}{R_e} \,\,\,\, \text{solve for }R_2 $$ when the teacher said "What you start out with is ...
1
vote
3answers
69 views

Difference of squares

Determine if $5^{36} - 1$ is divisible by $13$ using the difference of squares. I tried splitting it up by difference of squares a couple of times but can't seem to get to a point where I can ...
6
votes
2answers
286 views

struggle simplifying $\sqrt{9+\sqrt{5}}$

I need to simplify $\sqrt{9+\sqrt{5}}$ I already do this (proven it) $\sqrt{9-4\sqrt{5}}=2- \sqrt{5}$ But I couldn't when apply to ...
0
votes
1answer
51 views

when will they meet?

Suppose two runners are 1 km apart and they start running towards each other. Runner A starts running at such a speed that he would run a kilometer in 20 minutes. Runner B starts running at such a ...
3
votes
3answers
676 views

simplify $\sqrt{3+2\sqrt{2}}-\sqrt{4-2\sqrt{3}}$

Simplify $\sqrt{3+2\sqrt{2}}-\sqrt{4-2\sqrt{3}}$ To do it I have see it that we have basically $\sqrt{2}$ and $\sqrt{3}$ that is we can write it as, ...
3
votes
3answers
132 views

Best way to discover the 'golden ratio'

Let's say you live in a world where nobody ever discovered the Golden ratio. What's the most intuitive way to discover this proportion? Wikipedia defined it this way: $$\phi = \frac{a+b}{a} = ...
0
votes
2answers
1k views

Find Domain and Range of Composite Function

Given $$f(x) = \{(-5,0),(-4,-2),(-2,3),(1,5),(4,2)\}$$ $$g(x) = \{(0,-2),(8,4),(-2,5),(5,-5),(3,1)\}$$ $1$) Find the domain and range of $f(g(x))$. $2$) Find the domain and range of $g(f(x)$.
1
vote
1answer
45 views

If $z\in \mathbb{C}.$ Then minimum value of $\left|z^2-z+1\right|+\left|z^2+z+1\right|$

(1) If $\left|z\right| = 1$. Then find minimum value of $\left|z^2+z+4\right|$ (2) If $z\in \mathbb{C}.$ Then minimum value of $\left|z^2-z+1\right|+\left|z^2+z+1\right|.$ $\bf{My\; Try}::$ (1) ...
2
votes
1answer
96 views

A “complex” complex number problem

$a,b,c$ are cube roots of $p$ ,($p<0$) then for any permissible value of $x,y,z$ which is given by $$\frac{|xa+yb+zc|}{|xb+yc+za|} + (a_1^2-2b_1^2)\omega + \omega^2([x]+[y]+[z]) = 0 $$ $\omega$ ...
3
votes
4answers
371 views

Solve $2^{x}=x^{2}$

I've been asked to solve this and I've tried a few things but I have trouble eliminating x. I first tried taking the natural log: $x\ln \left( 2\right) =2\ln \left( x\right) $ $\dfrac {\ln \left( ...
0
votes
2answers
73 views

Basic math question

I'm going back to school and haven't taken a math class in years, so I'm brushing up on the basics. The text states $\frac{g(t + \Delta(t))^2}{2} = \frac{gt^2}{2} + \frac{g}{2}\left(2t\Delta t + ...
3
votes
2answers
259 views

Proving $p\nmid \dbinom{p^rm}{p^r}$ where $p\nmid m$

A question from Advanced Modern Algebra by Joseph J.Rotman. Let $n=(p^r)m $ such that the prime $p\nmid m$.Prove that $p\nmid \dbinom{n}{p^r}$.HINT: Assume otherwise,cross multiply and apply ...
1
vote
0answers
28 views

Finding the gradient from two points

I've come across the following question in a text book: Find the gradient of the line joining the following pair of points: (p+3, q-7), (p+5, 3-q) So I did the following: y2-y1 -> 3-q-q+7 ...
2
votes
8answers
528 views

Simplify $2^{(n-1)} + 2^{(n-2)} + … + 2 + 1$

Simplify $2^{(n-1)} + 2^{(n-2)} + .... + 2 + 1$ I know the answer is $2^n - 1$, but how to simplify it?
0
votes
2answers
102 views

What is my overall grade percentage score?

I received 20/30 for one assignment worth 30% of the overall mark. 21/30 for the second, which is worth 30% of the overall mark. And 15.5/40 for the third which is worth 40%. What is my overall grade ...
2
votes
2answers
46 views

Finding $x$. The summation of the floor of the equation.

I would appreciate if somebody could help me with the following problem Q:Finding $x$. The summation of the floor of the equation. $$\sum_{i=1}^{2013}\left\lfloor\frac{x}{i!}\right\rfloor=1001$$
0
votes
2answers
36 views

Percentages and Proportions

Given two integer variables $x$ and $y$. We are given that each integer variable $x$ and $y$ can't be greater than a given integer $z$. The problem: We are given the proportions $a$ and $b$ such that ...
1
vote
4answers
61 views

Integer $a$ , If $x ^ 2 + (a-6) x + a = 0 (a ≠ 0)$ has two integer roots

If the equation $x ^ 2 + (a-6) x + a = 0 (a ≠ 0)$ has two integer roots. Then the integer value of $a$ is $\bf{My\; Try}::$ Let $\alpha,\beta\in \mathbb{Z}$ be the roots of the equation . Then ...
0
votes
4answers
622 views

2 Easy GRE questions

I've been having trouble with these two questions. The first is simple interest, the second is rate. I'm sure they're easy but I can't focus on getting the solution because I'm terrible at focusing on ...
7
votes
3answers
112 views

Solve the equation $\lfloor x^2\rfloor-3\lfloor x \rfloor +2=0$

Solve the equation $$ \lfloor x^2\rfloor-3\lfloor x \rfloor +2=0 $$ where $\lfloor x\rfloor $ denotes floor function. My Attempt: Let $x = n+f$, where $n= \lfloor x \rfloor \in ...