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
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2answers
34 views

Show that if $a,\,b\in \mathbb{R}$ then [closed]

Show that if $a,\,b\in \mathbb{R}$, then $\min{(a,b,c)} = \min{(\min{(a,b)},c)}$ I have no clue where to even begin how to start this problem. I am so lost, I really need some help on this.
0
votes
0answers
8 views

a) Model the data with a linear function using the points in years 2000 and 2010.

The following data represents a company’s revenue in millions of dollars. Year: 2000 2002 2004 2005 2007 2008 2009 2010 2013 Revenue: 30 32 34 35 ...
0
votes
3answers
26 views

Looking to find the slope of a tangent line.

$$F(x)=x^3-3x^2-1$$ Find the slope of tangent line at $x=2$. Find the equation of the tangent line at $x=2$. Graph $f(x)$ and the tangent line on the same graph.
0
votes
2answers
43 views

How do I solve a multivariable equation?

How could I solve for variables $x$, $y$, $z$, and $w$ for the equation $$ax+by+cz+dw$$ With given values $a$, $b$, $c$, and $d$. For example, how would I find a set of potential values for $x$, ...
1
vote
1answer
71 views

Find summation of following series.

What will be the formula for following infinite series? $$1 + \frac{1!}{x+1} + \frac{2!}{(x+1)(x+2)}+ \cdots$$ $$ x\ge2 $$ up to infinite What pattern i got : coefficient of $ \frac{1!}{x+1}$ ...
0
votes
3answers
41 views

Prove that for positive $a,b,c : a^2+b^2=c^2\Rightarrow a+b < c \sqrt 2.$

Prove that for positive $a,b,c : a^2+b^2=c^2\Rightarrow a+b < c \sqrt 2.$ Is it solved considering a right isosceles triangle? I'm stuck on it
0
votes
3answers
49 views

How to simplify $\sqrt[3]{29\sqrt{2}-45}-\sqrt[3]{29\sqrt{2}+45}$

I in trouble simplifying this: $$\sqrt[3]{29\sqrt{2}-45}-\sqrt[3]{29\sqrt{2}+45}$$ couldn't find a solution. Can you help?
0
votes
3answers
76 views

$(x+1)/x = \sqrt{3}$ in form $a+b \sqrt{3}$

The above is the problem. I have trivially arrived at $$ x=\frac{1}{\sqrt{3}-1} $$ However I am unsure of how to get it into the required format. Thanks
0
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0answers
69 views
+50

$x^3+b^2x^2+2x+3=0$,Find several integer values of b such that the equation has roots.

$x^3+b^2x^2+2x+3=0$, Find several integer values of $b$ such that the equation has roots. My solution: I use the rational root theorem. $-3,3$ can be its rational roots. $$P(x)=x^3+b^2x^2+2x+3$$ ...
0
votes
1answer
37 views

Need help with simplifying a radical expression

I need help with simplifying this radical expression: $\sqrt{(5+2\sqrt{6})}(49-20\sqrt{6})(9\sqrt{3}+11\sqrt{2})$.
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votes
2answers
67 views

unclear transformation in identity proof from a textbook

We have $a+b+c=\pi$ and are given the task to prove that $$\sin(a)+\sin(b)+\sin(c)=4\cos\left(\frac a2\right)\cos\left(\frac b2\right)\cos\left(\frac c2\right)$$ The first steps of the proof are ...
22
votes
6answers
2k views

A beautiful game of gold and silver coins

A stack of silver coins is on the table. For each step we can either remove a silver coin and write the number of gold coins on a piece of paper, or we can add a gold coin and write the number of ...
2
votes
3answers
81 views

Problem about right triangles.

Given N>1 right triangles. Sum one legs of each of them, then sum all the left legs, then sum all hypotenuses. These 3 sums form the sides of a right triangle. Prove all given N triangles are similar ...
0
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3answers
54 views

Problem about sets of integers

Given 15 pairwisely different integers. Pat wrote all sums of 7 integers and Vova wrote all sums of 8 integers from this set. Can the set of sums of Pat be equivalent to the set of sums of Vova? I'm ...
3
votes
1answer
169 views

I get two different answers on simple equation. What am I doing wrong?

For the equation: $-x^2 = -2x(3x+1)$ I can either multiply it out on the right side and get a $-6x^2-2x$ or just divide both sides by $-2x$. However, when divide out both sides, I just get one answer: ...
1
vote
1answer
24 views

Adjusting weight of a body of water by substituting part of it with a lighter liquid

As a heavily simplified example of my problem: Water weighs 1 gram per ML Alcohol weighs 0.5 gram per ML (not true of course, but humour me) I have 100mls of water, so this has a weight of ...
1
vote
2answers
25 views

Meal Platters Optimization Problem

Mark has to buy hamburgers, hot dogs, and pig's feet for an event. The restaurant he is purchasing from offers two Platter options. Platter A comes with 4 hamburgers, 3 hot dogs, and 2 pig's feet. ...
0
votes
1answer
40 views

Calculate $X$ of a math problem

I am trying to learn some more math and I got stuck on this: $$\frac{0.2}{X} = 140$$ How do I calculate $X$? EDIT Sorry I meant to calculate $$\frac{28}{X} = 140$$ So that $X = 0.2$, but how do I ...
0
votes
1answer
69 views

Writing in closed form this nasty expansion

hi I have to write the following in closed form, $$a_1 + a_2 + a_3 + a_1 v_{2}v_{1}+ a_1v_{3}v_{1}+ a_2v_{3}v_{2}+a_2v_{1}v_{2}+ a_3v_{1}v_{3}+a_3v_{2}v_{3}$$ $$ \sum_{(i,j)\in \mathcal{S}} a_i(1+ ...
0
votes
1answer
37 views

Simplify a parametric equation with hyperbolic trigonometric functions

I've the following parametric equations for a curve: $$\begin{cases}x(t)=a\cdot \operatorname{sech} (t) \\ y(t)=a\cdot(t-\tanh(t))\end{cases}$$ Now let $\theta(t)=-\arctan(\sinh(t))$ how does the ...
1
vote
1answer
32 views

Find a function where the mode is the minimum

Let $a_i\in\Bbb R$ some collection of data points where $0\le i\le n$. Define the function $$f(x)=\sum_{i=0}^n(x-a_i)^2$$ It is clear that the minimum value of $f$ occurs when $x$ is the mean of ...
1
vote
3answers
147 views

Find the distance between two towns given train timings

While practicing maths and starting to learning it, I found question this question: A train running between two towns arrives at its destination 10 minutes late when it goes 40 miles per hour and ...
0
votes
1answer
48 views

Simplifying the quotient $\frac{4x^4+2x^2+x+1}{x^2+1}$

I got stuck simplifying the following quotient. How to divide it? $$\frac{4x^4+2x^2+x+1}{x^2+1}$$ Thanks a lot!
0
votes
2answers
46 views

Find the roots of the quadratics function

A problem on Khan Academy that I can't seem to wrap my head around... $$-3x^{ 2 }=x-6$$ This is how I manipulated the equation... $$3x^{ 2 }+x-6=0$$ This is how Sal did... $$-3x^{ 2 }-x+6=0$$ ...
0
votes
1answer
36 views

182-Day T-Bill vs. 91-Day T-Bill

I am trying to understand how T-Bills work and it would be great if someone could explain me using the following question At $t=0$ Smith buys a 182-Day T-Bill with a simple annual discount rate of ...
0
votes
2answers
48 views

Where do I start learning and how do I stay motivated?

I just finished intermediate algebra in college. I liked it and breezed through it. I feel like at this level of math I can only expect dull and unenthusiastic teachers (which has been the case ...
0
votes
3answers
40 views

Find the $\lim_{x\to -6} \frac{2x+12}{\lvert x+6 \rvert}\ $

Find the Find the $\lim_{x \to -6}\frac{2x+12}{|x+6|}\ $ The limit changes on when -6 approaches on the negative side to -(x-6) which gives me a denominator in the bottom. What have I done wrong?
3
votes
3answers
46 views

Growth of fraction of products with $\sqrt{n}$ terms

Is the growth of $$f(n):=\dfrac{(n+1)(n+2)\ldots(n+\sqrt{n})}{(n-1)(n-2)\ldots(n-\sqrt{n})}$$ polynomial or not? That is, does there exist constants $k,m$ such that $$f(n)<n^k$$ for all $n>m$?
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5answers
2k views

Prove that the sum of three consecutive squares, minus two is a multiple of 3

Prove that if you add the squares of three consecutive integer numbers and then subtract two, you always get a multiple of 3.
1
vote
1answer
41 views

Show that inequality is correct for natural $n$

Show that the following inequality is correct for all natural $n$ : $$(2n+1)^n\geq(2n)^n+(2n-1)^n$$ I've tried throwing the $(2n-1)^n$ or $(2n)^n$ on the left side and using formula of subtraction ...
1
vote
1answer
50 views

On nested circles

I have a problem with recursive formulas. I appreciate everyone can help me. The problem: We have n circles within each other. I want to achieve the value of radius of each circle such that the area ...
2
votes
3answers
62 views

How to solve $10^{x^2+x}+\log{x} = 10^{x+1}$?

In one of my recent exam, I was ask to solve this: $$ 10^{x^2+x}+\log{x} = 10^{x+1} $$ My attempt to solve it was: $$ 10^{x^2+x}+\log{x} = 10^{x+1} \\ \log{x}=10^{x+1}-10^{x^2+x} \\ ...
0
votes
0answers
53 views

Find factors of $0.08x^3 - 3.84x^2 + 42.66x - 137.7625$ using the Cubic Formula.

I have been going over this page as of late learning how to solve cubic formulas through depressing the equation, and solving for 'X'. Though, so far through numerous attempts, every single root I ...
3
votes
2answers
43 views

How do I evaluate $\lim_{x \to -1} \frac {x^2+2x+1}{x^2+4}$?

I have determined so far that this is equal to $$\lim_{x \to -1} \frac {(x+1)(x+1)}{(x+\sqrt [4] {1})(x-\sqrt [4]{1})(x^2+\sqrt{1})}.$$ However, my numerator becomes $0$ if I substitute the limit. ...
0
votes
1answer
33 views

How do you expand this $ \ \left(t-\sqrt[4]{2}\right)\left(t+\sqrt[4]{2}\right)\left(t^2+\sqrt 2\right) \ $

$ \ \left(t-\sqrt[4]{2}\right)\left(t+\sqrt[4]{2}\right)\left(t^2+\sqrt 2\right) \ $ I keeping on getting $ \ t^4+t\sqrt{2}+2t^2+2\sqrt{2} \ $
0
votes
2answers
33 views

Horizontal Asymptote of Strange Function

What is the horizontal asymptote as x approaches positive infinity of $\sqrt{4x^2 + 5x} - \sqrt{4x^2 + x}$? The horizontal asymptote is in the form $y = k$. Find $k$.
1
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0answers
18 views

Finding the highest power of N to fit in a given power of 2

I am trying to find the highest power $p$ of a number $N$ that will fit in a given power of 2. To give this some context, I am trying to find the largest power of $N$ that will fit in a 64-bit signed ...
0
votes
3answers
49 views

How do I factor $\ t^4-2 \ $?

This binomial is part of a bigger problem that I need to solve, however, I am little stuck on how to factor it. $(t-1)(t-1)(t+1)$ does not work.
0
votes
2answers
28 views

Simplify a trigonometric equation in quadratic form

I have a computer problem that I was able to reduce to an equation in quadratic form, and thus I can solve the problem, but it's a little messy. I was just wondering if anybody sees any tricks to ...
0
votes
4answers
36 views

Equation Subject to condition

Given that $ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 0 $, then prove that the following is true $$ \frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = - \frac{a^3 + b^3 + c^3}{abc} $$
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0answers
18 views

In an equation of a Hyperbola, what is the relation of the a and b terms?

I am studying hyperbolic equations (with conic sections) in pre-calculus algebra. I am a bit confused about the order of the a and b terms in the equation for a hyperbola with center at (h, k): ...
1
vote
2answers
46 views

Algebra: How to solve for a variable [closed]

Can someone help me solve for these variables please? And show the steps on how they did it. If: $$\frac{\left( \frac{x}{y} \right)}{1+\left( \frac{x}{y} \right)} = z$$ Solve for x. i.e. x= Solve ...
3
votes
2answers
41 views

Inequality proof of integers

My question is from Apostol's Vol. 1 One-variable calculus with introduction to linear algebra textbook. Page 36. Exercise 7. Let $n_1$ be the smallest positive integer $n$ for witch the inequality ...
2
votes
3answers
92 views

Prove that $2\sqrt{n}\sqrt{n+1} < 2n + 1$ for all positive integers.

I've been testing this with many values and it seems to always be true. I've been trying to rework the inequality into a form where it's much more obvious that the left hand side is always less than ...
0
votes
3answers
32 views

Canonical decompositions and product of primes

Let $S$ be the set of natural numbers $n$ that have exactly $9$ positive divisors. Describe all possible canonical decompositions (as products of primes) of elements of $S$.
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0answers
20 views

How to get a fractional exponent in the denominator to become a whole number exponent.

2x^-7/4 over 4x^4/3 *Simplify so that there are no negative exponents and no fractional exponents in the denominator.
0
votes
0answers
30 views

How to solve radical equations without creating any extraneous solutions while solving.

How does one solve a radical equation, such as $\sqrt{z+c_1}=z+c_2$, where $z \in \mathbb{C}, $ without creating any extraneous solutions at all while solving the equation. I know that by squaring an ...
0
votes
1answer
35 views

Factor Theorem given two factors

The function $f(x)= ax^3-x^2+bx-24$ has three factors. Two of these factors are $x-2$ and $x+4$. Determine the values of a and b and then solve for $f(x)$. Please give an algebraic solution.
0
votes
2answers
79 views

Find all natural number solutions to: $20x^2 + 11y^2 = 2011$

I believe that the equation $20x^2 + 11y^2 = 2011$ describes an ellipse. I don't know how to solve for the $x,y \in \mathbb{N}$ that satisfy this equation.
1
vote
2answers
98 views

How to show that $a,\ b\in {\mathbb Q},\ a^2+b^2=1\Rightarrow a=\frac{s^2-t^2}{s^2+t^2},\ b= \frac{2st}{s^2+t^2} $

I want show the following $$a,\ b\in {\mathbb Q},\ a^2+b^2=1\Rightarrow a=\frac{s^2-t^2}{s^2+t^2},\ b= \frac{2st}{s^2+t^2},\ s,\ t\in{\mathbb Q} $$ How can we prove this ? [Add] Someone implies that ...