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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2answers
71 views

Question about the imaginary unit [duplicate]

As we know, we define $$\sqrt{-1}=i$$ But I always wondered, what about $\sqrt{i}$? As far as i can see, it is not an integer power of $i$. Every odd root has a solution in an integer power of $i$, ...
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1answer
84 views

How to have the best solution for this equation $4\sqrt{x^5}-10x^2-(5-8\sqrt{x})x+\sqrt{x}=0$

How to have the best solution for this equation $4\sqrt{x^5}-10x^2-(5-8\sqrt{x})x+\sqrt{x}=0$ (1) I set $t=\sqrt{x}$ and $(1) \Leftrightarrow t(4t^4-10t^3-5t^2+8t+1)=0$ So It's difficult :( Thank ...
0
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1answer
127 views

Determine all the values of the parameter $a$ for which the inequality $3-|x-a|>x^2$ is satisfied by at least one negative $x$.

I wanted to know, how can I determine all the values of the parameter $a$ for which the inequality $3 - |x-a| > x^2$ is satisfied by at least one negative $x$. I tried for $x<a, |x-a|=-(x-a)$ ...
67
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13answers
5k views

What would have been our number system if humans had more than 10 fingers? Try to solve this puzzle.

Try to solve this puzzle: The first expedition to Mars found only the ruins of a civilization. From the artifacts and pictures, the explorers deduced that the creatures who produced this ...
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1answer
157 views

Arctanh to exp: Prove two equations are equivalent

For some peace of mind in a project, I am trying to prove two equations are somewhat equivalent. I have these two equations. $$ i_{1} = ...
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4answers
111 views

How to find the value of $a$ for which $\;\tan^2x + (a+1)\tan x-(a-3)<0$ is true

I wanted to know, how can I find the value of $a$ for which the inequality $\tan^2x + (a+1)\tan x-(a-3)<0$ is true for at least one $x\in(0,\pi/2)$. I don't know how to proceed, any help is ...
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2answers
440 views

Maximum number of cans within a box

A box that is 4 ft. by 4 ft. by 4 ft. is packed with (cylindrical) cans that are 2 ft. high and have a diameter of 6 inches. When the box is fully packed with cans, how much space is wasted in the ...
3
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2answers
92 views

Question on radius problem

A car with 15 inch radius tires was driven on a trip of a distance equal to 400 miles. Two months later, with snow tires, the odometer indicated 390 miles for the same trip. Find the radius of the ...
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1answer
73 views

Exponentials, Logarithms & the Natural Log

Could someone show me how to solve this problem? I don't know what "\" in front of "ln" means either. Solve the following equation for $y = f(x)$. $$e^y = e^{2y}e^{\ln(2x)}$$
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1answer
657 views

Folding a paper in half - Crease Lines

A strip of paper is folded in half, then the result is folded in half again, and the process is repeated for a total of 6 times (including the first fold). How many creases (fold lines) are there?
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1answer
400 views

Is $\large \frac {\pi}{e}$ rational, irrational, or trandescendal?

Is there an argument for why $\large \frac {\pi}{e}$ is rational, irrational, or trandescendal? Can the quotient of any two transcendental numbers (which are not rational multiples of each other) be ...
4
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6answers
169 views

Calculate the integer part

I have to calculate the integer part of this: $$[(\sqrt{2}+\sqrt{5})^2]$$ I tried to write it like this: $$[2+5+2\sqrt{10}]=[7+2\sqrt{10}]=7+[2\sqrt{10}]$$ Any ideas?
2
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2answers
530 views

If an AP, a GP and a HP have the same first term and same $(2n+1)$th terms and their $n$th terms are $a,b,c$ respectively

If an arithmetic preogression, a geometric progression and a harmonic preogression have the same first term and same $(2n+1)$th terms and their $n$th terms are $a,b,c$ respectively, then find the ...
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1answer
98 views

English language trick in mathematics problem

i would like to consider following problem,which mostly became trick because of hidden information what is actually asked,when i was preparing for GRE questions ,there was asked such kind of ...
2
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3answers
149 views

Inequalities - Absolute Value $|2x-1| \leq |x-3|$

$$|2x-1| \leq |x-3|$$ Answer is $$-2 \leq x \leq \frac43$$ My Question is HOW?
3
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1answer
305 views

how to find the roots of a cubic equation?

Given a formula $$x^3+ax^2+bx+c=0$$ how can I get the value of x without having an $i$ in my roots? Because Cardano's formula does have imaginary numbers if the discriminant is less than zero. My ...
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6answers
860 views

Average of all 6 digit numbers that contain only digits $1,2,3,4,5$

How do I find the average of all $6$ digit numbers which consist of only digits $1,2,3,4$ and $5$? Do I have to list all the possible numbers and then divide the sum by the count? There has to be a ...
0
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2answers
224 views

Finding the values of x for an equation

Find x in Z from this inequality: $$\frac32\left|x-\frac32\right|=\frac53|2x|-\frac16$$ I tried to solve it,but i don't know how to continue: $$\frac32\left|2x-\frac32\right|=\frac103|x|-\frac16$$ ...
4
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1answer
81 views

Figure out domain and range

Is there a hard and fast way, step by step process to figure out domain and range? I don't know where to start and lack the insight to just know what it can and can't be. Thanks
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2answers
67 views

Finding $x$ from inequality: $\left | \frac{3^n + 2}{3^n + 1} - 1 \right | \le \frac{1}{28}$

Find $x$ in $\mathbb{Z}$ satisfying this inequality: $$\left | \frac{3^n + 2}{3^n + 1} - 1 \right | \le \frac{1}{28}.$$ I tried something, but I don't think it's correct. $$-\frac{1}{28} ...
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2answers
28 views

If $D=\{(x,y):|x|+|y|\leq 1-z^4\}$ then $D$ is a square with side $\sqrt{2}(1-z^4)$?

If $D=\{(x,y):|x|+|y|\leq 1-z^4\}$ then $D$ is a square with side $\sqrt{2}(1-z^4)$. The absolute values can be interpreted as lengths with respect to their respective axes, but I still don't get it. ...
3
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1answer
1k views

Time between 3 and 4 that the hour and minute hands overlap each other?

This problem is on the practice problems for my math club, but I'm not sure how to solve it. What's the exact time that the hour and minute hands overlap each other between 3 and 4 o clock on an ...
3
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2answers
119 views

Finding all solutions to an inequality equation

I have the following inequality that I need to find all solutions of: $2x^3-8x > 5x^2-20$ My guess is that you would have to turn this into a polynomial equation and let the right hand side ...
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1answer
45 views

Points Calculator

I'm trying to formulate an equation for a online ranking system, I would like to award points to users based on there rank and the total number of users. I would like something similar to below, where ...
0
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1answer
719 views

Understanding the role of sales tax in problems involving prices

I would like to know what I should do in an algebraic calculation when there is sales tax mentioned. For example, let us suppose that the retail price of shirt is $R$ dollars, and the price including ...
2
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2answers
186 views

Comparing the “sizes” of square roots.

Let $\;x=3-\sqrt5,\;\;y=\sqrt5-2,\;\;z=5-2\cdot\sqrt5$. How can I tell without a calculator which is the largest and which is the smallest, in value?
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4answers
76 views

Percentage of balls that are not plastic nor gray in a jar

I am trying to revive my math skills if there are any :) but I can not come up with a solution to the following problem: You have a jar of balls. $55\%$ of the balls are gray, $45\%$ are plastic ...
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1answer
817 views

Trying to reverse an equation and solve for a different variable

My original question here produced this equation: $$\text{level} = \operatorname{int}\left(\left(\left(\sqrt{\text{xp}\times 8 + 100}\over10\right)-1\right)\div2\right)$$ Now I would like to reverse ...
0
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1answer
42 views

How to find the correct subdevision location given the total length, number of total subdevisions, and the location on that length?

I am working on a programing problem that I just cant seem to get the formula working correctly. Here is the problem: Given that a length L is subdivided ...
4
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6answers
697 views

Find the value of $x$ and $y$ given this equation

So I have a College Admission test tomorrow and I am hoping that you could help me understand how to arrive at the solution to this: 1.) Given the following equations: $$3x-y=30\\ 5x-3y=10$$ ...
1
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1answer
128 views

find the value of 1/(2+1/(4+1/(4+1/(…))))

the question is to find the value of this ugly non-stopping fraction $$\frac{1}{2+\frac{1}{4+\frac{1}{4+\frac{1}{\ldots}}}}$$. I have totally no clue; thanks for the help! How am I suppose to solve ...
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2answers
340 views

$\frac1a+\frac1b+\frac1c=0 \implies a^2+b^2+c^2=(a+b+c)^2$? [closed]

How to prove that $a^2+b^2+c^2=(a+b+c)^2$ given that $\frac1a+\frac1b+\frac1c=0$?
1
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2answers
140 views

Subadditivity inequality and power functions [duplicate]

Is it true that if $a,b\in\mathbb{R}$ with $a,b\geq 0$ and $0<r<1$, then $(a+b)^r\leq a^r+b^r$?
6
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4answers
330 views

How to show that a limit cannot be another number?

Let: $$ G(x) = \left\{ \begin{array} {cc} x \sin \frac{1}{x} , & x\neq 0 \\ 0, & x=0 \end{array} \right. $$ I can understand that the function is continuous at $x=0$ because: For ...
4
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3answers
165 views

how does this translate to a circle with radius 5: $\sqrt{24-2x-x^2}$

I tried squaring both sides to get this $y^2 = 24-2x-x^2$, then putting the $x$'s with the $y$'s to get $y^2 + x^2 + 2x = 24$. Then I tried dividing everything by 24, but I don't see it. Tried ...
3
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1answer
193 views

What maths is being used to calculate this interest?

I'm curious about how my bank is calculating the interest on my credit card. No matter what I do, I cannot make the numbers add up! Below is a photo of my latest statement. It's for 13th June - 12th ...
1
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1answer
922 views

Express spherical coordinates with different centers in terms of each other.

Imagine that you have two spheres with a distance $R$ from one center to the other one. Now, it is well known how one would get the cartesian position vector of each point in sphere 1 by using ...
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2answers
57 views

find any number from 6 equations

suppose i have six equations which are equal to different numerical values. x + y = A(numerical value) x + d = B(numerical value) y - d = C(numerical value) a + b = D(numerical value) a + d = ...
1
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1answer
98 views

cost of washing clothes

let us consider this problem: A drycleaner charges $2$ dollar for up to $3$ pounds in weight of clothes, and $30$ cents per pound or part thereof up to a maximum weight of $10$ pounds per load. What ...
3
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2answers
71 views

Simplify $(xy^{-1})/(y^2/x^3)^{-1}$

Need to simplify this equation: $$(xy^{-1})/(y^2/x^3)^{-1}$$ So far I have $= \frac{x/y}{1/(y^2/x^3)}$ But don't know what to do next. Any help much appreciated
2
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2answers
107 views

Whats wrong with this proof?

Theorem: $x$ is a real number with $x \neq 1.$ If $\frac {x^2+1}{x-1} =x$, then $x=-1$. If we suppose that $x=-1$. Then $\frac {x^2+1}{x-1} = \frac {(-1)^2+1}{-1-1} = \frac {2}{-2} = -1 = x$ I would ...
3
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3answers
2k views

Rounding number nearest 0.05

I have question about rounding and please help me, suppose that question is round given number nearest 0.01 or 0.1 or 0.05 or maybe nearest 0.5, then what could i do? For example we are given some ...
2
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3answers
230 views

Given that the roots of the quadratic equation $x^2+2ax+3a=0$ lie between $-1$ and $1$, what are the possible values of $a$?

In the equation $x^2+2ax+3a=0$ has two solutions $\alpha$ and $\beta$ where $-1<\alpha,\beta<1$. Find out the range of $a$. I tried to solve it by taking $\alpha^2+\beta^2$. But it does ...
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1answer
127 views

Finding the remainder .

How to find the remainder of $\dfrac{7^{8^9}}{1000}$
0
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2answers
76 views

How to find the sum of the series by treating deonominator so that to split fraction $\frac{1}{a_1a_2a_3} + \frac{1}{a_2a_3a_4}+$…

This is a series in A.P ( Arithmetic Progression ) $\frac{1}{a_1a_2} + \frac{1}{a_2a_3}+\frac{1}{a_3a_4}+.......\frac{1} { a_{n}a_{n+1}}$ ( where $a_1 ,a_2,a_3.....$ are terms in A.P.) When we do ...
2
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2answers
164 views

maximum using completing the square

Is it just me, or this problem does sound weird? The Parks Department is fencing a rectangular dog-run (a place for dogs to exercise) in a local park. It will be 7 yards longer than 5 times its ...
0
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1answer
157 views

compare slopes of two line

let us consider following problem: In the $xy$-plane, the point $(1, 2)$ is on line $j$, and the point $(2, 1)$ is on line $k$. Each of the lines has a positive slope. we should compare slopes ...
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2answers
1k views

Proof the maximum function $\max(x,y) = \frac {x +y +|x-y|} {2}$ [duplicate]

I want to prove the maximum function max: $\mathbb{R} \rightarrow \mathbb{R}$, which is defined by $$\max(x,y) = \begin{cases}x, \text { if } x \geq y , \\ y, \text { if } x < y \end{cases}$$ ...
1
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1answer
689 views

Sketching a Cubic Polynomial?

Suppose that $f(x)$ has $(x-2)^2$ and $(x+1)$ as its only factors. How do I sketch the graph of $f$? So far what I've done is determine (hopefully correctly) that the x-intercepts will be at $-1$ and ...
29
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4answers
6k views

Are We Teaching Pre-Calc Wrong?

It took some 1,250 years to move from the integral of a quadratic to that of a fourth degree polynomial. When we jump too fast to the magical algorithm, when we fail to acknowledge the effort that ...